Sequence & Series : PAJ Sir
Sequence & Series Sequence : A sequence is a function whose domain is the set N of natural numbers. Since the domain for every sequence is the set N of natural numbers, therefore a sequence is represented by its range. If f : N R, then f(n) = tn, n N is called a sequence and is denoted by {f(1), f(2), f(3),...............} = {t1, t2, t3, ......................} = {tn}
Real sequence : A sequence whose range is a subset of R is called a real sequence. e.g.
(i) (ii) (iii)
2, 5, 8, 11, ....................... 4, 1, – 2, – 5, ...................... 3, –9, 27, – 81, ........................
Types of sequence :
(i) (ii)
On the basis of the number of terms there are two types of sequence. Finite sequences : A sequence is said to be finite if it has finite number of terms. Infinite sequences : A sequence is said to be infinite if it has infinitely many terms.
Example # 1 : Write down the sequence whose nth term is
Solution :
(i)
2n n
(i)
Let tn =
3 ( 1)n
(ii)
3n
2n n
put n = 1, 2, 3, 4, .............. we get t1 = 2, t2 = 2, t3 =
8 ,t =4 3 4
so the sequence is 2, 2,
(ii)
Let tn =
8 , 4, ........ 3
3 ( 1)n 3n
put n = 1, 2, 3, 4, ...... so the sequence is
2 4 4 2 , , , ,............ 3 9 27 81
Series : By adding or subtracting the terms of a sequence, we get an expression which is called a series. If a1, a2, a3,........an is a sequence, then the expression a1 + a2 + a3 + ...... + an is a series. e.g. (i) 1 + 2 + 3 + 4 + .................... + n (ii) 2 + 4 + 8 + 16 + ................. (iii) – 1 + 3 – 9 + 27 – ..............
RESONANCE
1
Sequence & Series : PAJ Sir
Progression : The word progression refers to sequence or series – finite or infinite
Arithmetic progression (A.P.) : A.P. is a sequence whose successive terms are obtained by adding a fixed number 'd' to the preceding terms. This fixed number 'd' is called the common difference. If a is the first term & d the common difference, then A.P. can be written as a, a + d, a + 2 d,....... ,a + (n 1) d,........ e.g.
– 4, – 1, 2, 5 ...........
n th term of an A.P. : Let 'a' be the first term and 'd' be the common difference of an A.P., then tn = a + (n – 1) d, where d = tn – tn – 1 Example # 2 : If t54 of an A.P. is – 61 and t4 = 64, find t10. Solution :
Let a be the first term and d be the common difference so t54 = a + 53d = – 61 .........(i) and t4 = a + 3d = 64 .........(ii) equation (i) – (ii) we get 50d = – 125 d=–
a=
5 2
143 2
so
t10 =
5 143 + 9 = 49 2 2
Example # 3 : Find the number of terms in the sequence 4, 12, 20, ........,108. Solution :
a = 4, d = 8 so n = 14
108 = 4 + (n – 1)8
The sum of first n terms of an A.P. : If a is first term and d is common difference, then sum of the first n terms of AP is Sn = =
n [2a + (n – 1) d] 2 n [a + ] nt n 1 , for n is odd. (Where is the last term and t n 1 is the middle term.) 2 2
2
Note : For any sequence {tn}, whose sum of first r terms is Sr, rth term, tr = Sr – Sr – 1.
Example # 4 : Find the sum of all natural numbers divisible by 5, but less than 100. Solution :
All those numbers are 5, 10, 15, 20, ..........., 95. Here a = 5, n = 19 & = 95 so
RESONANCE
S=
19 (5 + 95) = 950. 2
2
Sequence & Series : PAJ Sir Example # 5 : Find the sum of all the three digit natural numbers which on division by 7 leaves remainder 3. Solution :
All these numbers are 101, 108, 115, ........, 997 997 = 101 + (n – 1) 7 n = 129 so
S=
129 [101 + 997] = 70821. 2
Example # 6 : The sum of n terms of two A.Ps. are in ratio Solution :
7n 1 . Find the ratio of their 11 1th terms. 4n 27
Let a1 and a2 be the first terms and d1 and d2 be the common differences of two A.P.s respectively,
n [2a1 (n 1)d1 ] 7n 1 2 then n = 4n 27 [2a 2 (n 1)d2 ] 2
n 1 a1 d1 7n 1 2 = n 1 4n 27 a2 d2 2
For ratio of 11th terms
n 1 = 10 2
so ratio of 11th terms is =
n = 21
7(21) 1 4 148 = = 4(21) 27 3 111
Example # 7 : If sum of n terms of a sequence is given by Sn = 2n2 + 3n, find its 50th term. Solution :
Let tn is nth term of the sequence so tn = Sn – Sn – 1. = 2n2 + 3n – 2(n – 1)2 – 3(n – 1) = 4n + 1 so t50 = 201.
Self practice problems : (1)
Which term of the sequence 2005, 2000, 1995, 1990, 1985, ............. is the first negative term
(2)
For an A.P. show that tm + t2n + m = 2 tm + n
(3)
Find the maximum sum of the A.P. 40 + 38 + 36 + 34 + 32 + .............. Answers :
(1)
403
(3)
420
Remarks : (i)
The first term and common difference can be zero, positive or negative (or any complex number.)
(ii)
If a, b, c are in A.P. 2 b = a + c & if a, b, c, d are in A.P. a + d = b + c.
(iii)
Three numbers in A.P. can be taken as a d, a, a + d ; four numbers in A.P. can be taken as a 3d, a d, a + d, a + 3d; five numbers in A.P. are a 2d, a d, a, a + d, a + 2d ; six terms in A.P. are a 5d, a 3d, a d, a + d, a + 3d, a + 5d etc.
RESONANCE
3
Sequence & Series : PAJ Sir (iv)
The sum of the terms of an A.P. equidistant from the beginning & end is constant and equal to the sum of first & last terms.
(v)
Any term of an A.P. (except the first) is equal to half the sum of terms which are equidistant from it. an = 1/2 (ank + an+k), k < n. For k = 1, an = (1/2) (an1+ an+1); For k = 2, an = (1/2) (an2+ an+2) and so on.
(vi)
If each term of an A.P. is increased, decreased, multiplied or divided by the same non-zero number, then the resulting sequence is also an AP. The sum and difference of two AP's is an AP.
(vii)
Example # 8 : The sum of three numbers in A.P. is 27 and the sum of their squares is 293, find them. Solution :
Let the numbers be a – d, a, a + d so 3a = 27 a=9 Also (a – d)2 + a2 + (a + d) 2 = 293. 3a2 + 2d2 = 293 d2 = 25 d=±5 therefore numbers are 4, 9, 14. 5
Example # 9 : If a1, a2, a3, a4, a5 are in A.P. with common difference 0, then find the value of Solution :
a , when i
i1
a3 = 2. As a1, a2, a3, a4, a5 are in A.P., we have a1 + a5 = a2 + a4 = 2a3. 5
Hence
Example # 10 : If Solution :
i1
i
= 10.
1 1 1 , , are in A.P., prove that a2, b2, c 2 are also in A.P.. bc c a ab 1 1 1 , , are in A.P.. bc c a ab
1 1 1 1 – = – ca bc ab ca
bc c a c aa b = (c a)(b c ) (a b)(c a)
ba c b = bc ab
b2 – a2 = c 2 – b2 a2, b2, c 2 are in A.P.
Example # 11 : If
Solution :
a
bc a c ab abc 1 1 1 , , are in A.P., then prove that , , are also in A.P. a b c a b c
Given
bc a c ab abc , , are in A.P.. a b c
Add 2 to each term bc a c ab abc , , are in A.P.. a b c divide each by a + b + c
RESONANCE
1 1 1 , , are in A.P.. a b c
4
Sequence & Series : PAJ Sir
Arithmetic mean (mean or average) (A.M.) : If three terms are in A.P. then the middle term is called the A.M. between the other two, so if a, b, c are in A.P., b is A.M. of a & c. A.M. for any n numbers a1, a2,..., an is; A =
a 1 a 2 a 3 ..... a n . n
n - Arithmetic means between two numbers : If a, b are any two given numbers & a, A1, A2,...., An, b are in A.P., then A1, A2,... An are the n A.M.’s between a & b. A1 = a +
n (b a ) ba 2 (b a ) , A2 = a + ,......, An = a + n 1 n1 n1
Note : Sum of n A.M.’s inserted between a & b is equal to n times the single A.M. between a & b n
i.e.
r 1
Ar = nA, where A is the single A.M. between a & b i.e.
A=
ab 2
13 , an even number of A.M.s is inserted, the sum of 6 these means exceeds their number by unity. Find the number of means.
Example # 12 : Between two numbers whose sum is
Solution :
Let a and b be two numbers and 2n A.M.s are inserted between a and b, then 2n (a + b) = 2n + 1. 2 13 13 n = 2n + 1. given a b 6 6 n = 6. Number of means = 12.
Example # 13 : Insert 20 A.M. between 2 and 86. Solution : Here 2 is the first term and 86 is the 22nd term of A.P. so 86 = 2 + (21)d d=4 so the series is 2, 6, 10, 14,......., 82, 86 required means are 6, 10, 14,...,82. Self practice problems : (4)
If A.M. between pth and qth terms of an A.P. be equal to the A.M. between r th and s th terms of the A.P., then prove that p + q = r + s.
(5)
If n A.M.s are inserted between 20 and 80 such that first mean : last mean = 1 : 3, find n.
(6)
For what value of n,
Answers :
(5)
a n1 b n 1 an bn
n = 11
, a b is the A.M. of a and b. (6)
n=0
Geometric progression (G.P.) : G.P. is a sequence of numbers whose first term is non zero & each of the succeeding terms is equal to the preceeding terms multiplied by a constant. Thus in a G.P. the ratio of successive terms is constant. This constant factor is called the common ratio of the series & is obtained by dividing any term by that which immediately preceeds it. Therefore a, ar, ar 2, ar3, ar4,...... is a G.P. with 'a' as the first term & 'r' as common ratio. e.g. (i) 2, 4, 8, 16, .......
RESONANCE
5
Sequence & Series : PAJ Sir
Results :
(ii)
1 , 1 , 1 , 1 ,....... 3 9 27 81
(i) (ii)
nth term of GP = a rn1 Sum of the first n terms of GP
a rn 1 Sn = r 1 na
(iii)
, r 1 , r 1
Sum of an infinite terms of GP when r < 1. When n rn 0 if r < 1 therefore, S =
a r 1 . 1 r
Example # 14 : If the first term of G.P. is 7, its nth term is 448 and sum of first n terms is 889, then find the fifth term of G.P. Solution :
Given a = 7 tn = arn – 1 = 7(r)n – 1= 448. 7rn = 448 r Also
Sn =
a(r n 1) 7(r n 1) = r 1 r 1
448r 7 r 1 Hence T 5 = ar4 = 7(2)4 = 112.
889 =
r=2
Example # 15 : The first term of an infinite G.P. is 1 and any term is equal to the sum of all the succeeding terms. Find the series. Solution : Let the G.P. be 1, r, r2, r3, .........
given condition
Hence series is 1,
Example # 16 : Let S = 1 + (i)
Solution :
(i)
r=
r2 1 r
r=
1 , 2
1 1 1 , , , .............. 2 4 8
1 1 1 + + + .........., find the sum of 8 2 4
first 20 terms of the series
S20 =
1 20 1 2 1 1 2
=
2 20 1 19
2
(ii)
.
infinite terms of the series.
(ii)
S =
1 1
1 = 2. 2
Self practice problems : (7)
Find the G.P. if the common ratio of G.P. is 3, nth term is 486 and sum of first n terms is 728.
(8) (9)
If the pth, qth, rth terms of a G.P. be a, b, c respectively, prove that aq – r br – p c p – q = 1. A G.P. consist of 2n terms. If the sum of the terms occupying the odd places is S1 and that of
RESONANCE
6
Sequence & Series : PAJ Sir the terms occupying the even places is S2 , then find the common ratio of the progression. (10)
The sum of infinite number of terms of a G.P. is 4, and the sum of their cubes is 192, find the series.
Answers :
(7)
2, 6, 18, 54, 162, 486
S2 . S1
(9)
(10)
6, – 3,
3 ,........ 2
Remarks : (i)
If a, b, c are in G.P. b2 = ac, in general if a1, a2, a3, a4,......... an – 1 , an are in G.P., then a1an = a2an – 1 = a3 an – 2 = ..........................
(ii)
Any three consecutive terms of a G.P. can be taken as
(iii)
Any four consecutive terms of a G.P. can be taken as
(iv)
If each term of a G.P. be multiplied or divided or raised to power by the same nonzero quantity, the resulting sequence is also a G.P..
(v)
If a1, a2, a3,........ and b1, b2, b3,......... are two G.P’s with common ratio r 1 and r2 respectively, then the sequence a1b1, a2b2, a3b3, ..... is also a G.P. with common ratio r1 r2.
(vi)
If a1, a2, a3,..........are in G.P. where each ai > 0, then log a1, loga2, loga3,..........are in A.P. and its converse is also true.
a , a , ar.. r a
r3
,
a , ar, ar3. r
Example # 17 : Find three numbers in G.P. having sum 19 and product 216. a Solution : Let the three numbers be , a, ar r 1 so a 1 r = 19 .......(i) r and a3 = 216 a=6 so from (i) 6r2 – 13r + 6 = 0. 3 2 r= , 2 3 Hence the three numbers are 4, 6, 9. Example # 18 : Find the product of 11 terms in G.P. whose 6th term is 5. Solution :
Using the property a1a11 = a2a10 = a3a9 = .............. = a62 = 25 Hence product of terms = 511
Example # 19 : Using G.P. express 0. 3 and 1.2 3 as Solution :
Let
p form. q
x = 0. 3 = 0.3333 ............. = 0.3 + 0.03 + 0.003 + 0.0003 + ............. =
3 3 3 3 + + + + .............. 10 100 1000 10000
3 3 1 10 = = = . 1 9 3 1 10
RESONANCE
7
Sequence & Series : PAJ Sir Let
y = 1.2 3 = 1.233333 = 1.2 + 0.03 + 0.003 + 0.0003 + ............. = 1.2 +
3 10
2
+
3 10
3
+
3 10 4
+ ............
3 1 37 10 2 = 1.2 + = 1.2 + = . 1 30 30 1 10
Example # 20 : Evaluate 7 + 77 + 777 + ........... upto n terms. Solution :
Let S = 7 + 77 + 777 + ..........upto n terms. =
7 [9 + 99 + 999 + .......] 9
=
7 [(10 – 1) + (102 – 1) + (103 – 1) + ........ + upto n terms] 9
=
7 [10 + 102 + 103 + ...........+ 10n – n] 9
=
7 9
=
7 [10n + 1 – 9n – 10] 81
10(10n 1) n 9
Geometric means (mean proportional) (G.M.): If a, b, c are in G.P., b is called as the G.M. of a & c. If a and c are both positive, then b =
ac and if a and c are both negative, then b = –
ac .
b² = ac, therefore b = a c ; a > 0, c > 0.
n-Geometric means between a, b : If a, b are two given numbers & a, G 1, G2,....., Gn, b are in G.P.. Then G1, G2, G3,...., Gn are n G.M.s between a & b. G1 = a(b/a)1/n+1, G2 = a(b/a)2/n+1,......, G n = a(b/a)n/n+1 Note : The product of n G.M.s between a & b is equal to the nth power of the single G.M. between a & b n
i.e.
G = ab = G n
r 1
r
n
, where G is the single G.M. between a & b.
Example # 21 : Insert 4 G.M.s between 2 and 486. 1
Solution :
b n1 Common ratio of the series is given by r = = (243)1/5 = 3 a Hence four G.M.s are 6, 18, 54, 162.
RESONANCE
8
Sequence & Series : PAJ Sir Self practice problems : (11)
(12)
The sum of three numbers in G.P. is 70, if the two extremes be multiplied each by 4 and the mean by 5, the products are in A.P. Find the numbers.
111 .......... 1 If a = , b = 1 + 10 + 102 + 103 + 104 and c = 1 + 105 + 1010 + ..... + 1050, then prove 55
that Answers :
(i)
‘a’ is a composite number
(11)
10, 20, 40
(ii)
a = bc.
Harmonic progression (H.P.) A sequence is said to be in H.P if the reciprocals of its terms are in A.P.. If the sequence a1, a2, a3,...., an is in H.P. then 1/a1, 1/a2,...., 1/an is in A.P. Note : (i)
Here we do not have the formula for the sum of the n terms of an H.P.. For H.P. whose first term is a and second term is b, the nth term is tn =
ab . b (n 1)(a b)
a ab 2ac or = . ac c bc
(ii)
If a, b, c are in H.P. b =
(iii)
If a, b, c are in A.P.
a ab = a bc
(iv)
If a, b, c are in G.P.
a ab = b bc
Harmonic mean (H.M.): If a, b, c are in H.P., b is called as the H.M. between a & c, then b =
2ac ac
If a1, a2 , ........ an are ‘n’ non-zero numbers then H.M. 'H' of these numbers is given by
1 1 1 1 ....... 1 = an H n a1 a 2
Example # 22 : If m th term of H.P. is n, while nth term is m, find its (m + n) th term. Solution :
1 Given T m = n or a (m 1) d = n; where a is the first term and d is the common difference of the corresponding A.P. so
a + (m – 1)d =
1 n
and
a + (n – 1) d =
1 m
or
d=
1 mn
Hence T (m + n) =
RESONANCE
(m – n)d =
so
a=
mn mn
1 (m 1) 1 – = n mn mn
1 mn mn = = . a (m n 1) d 1 m n 1 mn
9
Sequence & Series : PAJ Sir 2 2 and . 3 13
Example # 23 : Insert 4 H.M between Solution :
Let 'd' be the common difference of corresponding A.P..
so
13 3 d = 2 2 = 1. 5
1 3 5 H1 = 2 + 1 = 2
or
H1 =
2 5
1 3 7 H2 = 2 + 2 = 2
or
H2 =
2 7
1 3 9 H3 = 2 + 3 = 2
or
H3 =
2 9
1 3 11 H4 = 2 + 4 = 2
or
H4 =
2 . 11
Example # 24 : If pth, qth, rth terms of an H.P. be a, b, c respectively, prove that (q – r)bc + (r – p) ac + (p – q) ab = 0 Solution : Let 'x' be the first term and 'd' be the common difference of the corresponding A.P.. so
1 = x + (p – 1)d a
...........(i)
1 = x + (q – 1) d b 1 = x + (r – 1) d c
(i) - (ii) (ii) - (iii) (iii) - (i) (iv) + (v) + (vi) gives bc (q – r) + ac(r – p) +
..........(ii)
..........(iii)
ab(p – q)d = b – a bc (q – r)d = c – b ac (r – p) d = a – c
..........(iv) ..........(v) ..........(vi)
ab (p – q) = 0.
Self practice problems : (13)
If a, b, c be in H.P., show that a : a – b = a + c : a – c.
(14)
If the ratio of H.M. between two positive numbers 'a' and 'b' (a > b) is to their G.M. as 12 to 13, prove that a : b is 9 : 4.
(15)
If H be the harmonic mean of a and b, then find the value of
(16)
If a, b, c, d are in H.P., then show that ab + bc + cd = 3ad
Answers :
(15)
H H + – 1. 2a 2b
0
Relation between means : (i)
If A, G, H are respectively A.M., G.M., H.M. between a & b both being positive, then G² = AH (i.e. A, G, H are in G.P.) and A G H.
RESONANCE
10
Sequence & Series : PAJ Sir Example # 25 : The A.M. of two numbers exceeds the G.M. by Solution :
6 3 and the G.M. exceeds the H.M. by ; find 5 2
the numbers. Let the numbers be a and b, now using the relation G2 = AH 3 6 3 9 = G G = G2 + G– 2 5 10 5 G=6 i.e. ab = 36 also a + b = 15 Hence the two numbers are 3 and 12.
A.M. G.M. H.M. Let a1, a2, a3, .......an be n positive real numbers, then we define their a1 a 2 a 3 ....... a n A.M. = , their n 1/n G.M. = (a1 a2 a3 .........an) and their n H.M. = 1 1 1 . ....... a1 a 2 an It can be shown that A.M. G.M. H.M. and equality holds at either places iff a1 = a2 = a3 = ..............= an a b c + + 3 b c a Using the relation A.M. G.M. we have
Example # 26 : If a, b, c > 0, prove that Solution :
1 a b c 3 a b c b c a . . b c a 3
a b c 3 b c a
1 1 1 Example # 27 : If x,y,z are positive, then prove that (x + y + z) x y z 9 Solution :
Using the relation A.M. H.M.
3 xyz 1 1 1 3 x y z Example # 28 : If ai > 0 i N such that Solution :
Using A.M. G.M.
n
a
i
1 1 1 (x + y + z) x y z 9
1 , then prove that (1 + a1) (1 + a2) (1 + a3) ........(1 + an) 2n
i 1
1 + a1 2 a1 1 + a2 2 a 2
1 + an 2 an
(1 + a1) (1 + a2) .........(1 + an) 2n a1a 2 a 3 ......a n 1/ 2
As a1 a2 a3 ..... an = 1 Hence (1 + a1) (1 + a2) .......... (1 + an) 2n.
RESONANCE
11
Sequence & Series : PAJ Sir Example # 29 : If n > 0, prove that 2n > 1 + n 2n 1 Solution :
Using the relation A.M. G.M. on the numbers 1, 2, 22, 23,..........., 2n–1 , we have
1 2 22 ....... 2n1 > (1.2. 22. 23. ......... .2n–1)1/n n Equality does not hold as all the numbers are not equal. 1
(n 1) n n 2 2
2n 1 >n 2 1
2n – 1 > n. 2
2n > 1 + n. 2
(n 1) 2
(n 1) 2
Example # 30 : Find the greatest value of xyz for positive value of x, y, z subject to the condition xy + yz + zx = 12. Solution :
Using the relation A.M. G.M. xy yz zx (x2 y2 z2)1/3 3
4 (x y z)2/3
xyz 8
Example # 31 : If a, b, c are in H.P. and they are distinct and positive, then prove that an + cn > 2bn Solution :
Let an and cn be two numbers
an c n > (an cn)1/2 2 an + cn > 2 (ac)n/2 Also G.M. > H.M. then
i.e.
ac > b, (ac)n/2 > bn
...........(i) ...........(ii)
hence from (i) and (ii), we get an + cn > 2bn Self practice problems : (17)
If a, b, c are real and distinct, then show that a2 (1 + b2) + b2 (1 + c2) + c2 (1 + a2) > 6abc
(18)
Prove that nn 1 . 3 . 5 .........(2n – 1), n N
(19)
If a, b, c, d be four distinct positive quantities in G.P., then show that (i) a+d>b+c (ii)
1 1 1 1 1 + >2 bd ac ad ab cd
(20)
Prove that ABC is an equilateral triangle iff tan A + tan B + tan C = 3 3
(21)
If a, b, c > 0, prove that [(1 + a) (1 + b) (1 + c)]7 > 77 a4 b4 c4
Arithmetico-geometric series : A series, each term of which is formed by multiplying the corresponding terms of an A.P. & G.P. is called the Arithmetico-Geometric Series. e.g. 1 + 3x + 5x 2 + 7x 3 +..... Here 1, 3, 5,.... are in A.P. & 1, x, x 2, x 3..... are in G.P..
RESONANCE
12
Sequence & Series : PAJ Sir Sum of n terms of an arithmetico-geometric series: Let Sn = a + (a + d) r + (a + 2 d) r² +..... + [a + (n 1)d] rn1, then
d r 1 r n 1 a a (n 1) d r n , r 1. Sn = 1 r 1 r 1 r 2 Sum to infinity: If r < 1 & n , then Limit rn = 0 and Limit n.rn = 0 n n
S =
a dr . 1 r 1 r 2
Example # 32 : Find the sum of the series 1 + Solution :
Let
S=1+
7 10 4 + 2 + 3 + ...... to n terms. 5 5 5
7 10 3n 2 4 + 2 + 3 + ...... + 5 5 5 5n1
1 S= 5
..........(i)
4 7 3n 5 3n 2 1 + 2 + 3 + ....... + n1 + 5 5 5 5 5n
..........(ii)
(i) – (ii)
3 3 3 3n 2 4 3 S=1+ + 2 + 3 + ....... + n1 – . 5 5 5 5 5 5n 3 1 1 5 5 4 S =1+ 1 5 1 5
=
12n 7 7 – 4.5n 4
n1
–
3n 2 5n
S =
=1+
1 3n 2 3 3 – × n1 – 5 5n 4 4
(12n 7) 35 – . 16 . 5 n1 16
Example # 33 : Evaluate 1 + 2x + 3x 2 + 4x 3 + ......... upto infinity, where | x | < 1. Solution :
S = 1 + 2x + 3x 2 + 4x 3 + ..... xS= x + 2x 2 + 3x 3 + ......... (i) - (ii) (1 – x) S = 1 + x + x 2 + x 3 + .......... Let
........(i) ........(ii)
1 or
S =
(1 x )2
Example # 34 : Evaluate : 1 + (1 + b) r + (1 + b + b2) r2 + ......... to infinite terms for | br | < 1. Solution :
S = 1 + (1 + b)r + (1 + b + b2) r2 +..... rS = r + (1 + b) r 2 + ......... (i) - (ii) (1 – r)S = 1 + br + b2r2 + b3r3 + ...... Let
S=
..........(i) ..........(ii)
1 (1 br )(1 r )
Self practice problems : (22)
Evaluate : 1.2 + 2.22 + 3.23 + ...... + 100. 2100
(23)
Evaluate : 1 + 3x + 6x 2 + 10x 3 + ...... upto infinite term, where | x | < 1.
RESONANCE
13
Sequence & Series : PAJ Sir
(24)
1 1 Sum to n terms of the series : 1 + 2 1 + 3 1 n n
Answers :
+ ......
1
99.2101 + 2.
(22)
2
(23)
n2
(24)
(1 x )3
Results : n
(i)
n
(ar ± br) =
r 1
r 1
n
n
ar ±
b r.
r 1
(ii)
r 1
n
k ar = k
r 1
a r.
n
(iii)
k = k + k + k +...............n times = nk; where k is a constant.
r 1
n
(iv)
r = 1 + 2 + 3 +...........+ n =
r 1
n ( n 1) 2
r² = 12 + 22 + 32 +...........+ n2 =
n (n 1) (2n 1) 6
r3 = 13 + 23 + 33 +...........+ n3 =
n 2 (n 1) 2 4
n
(v)
r 1 n
(vi)
r 1
Example # 35 : Find the sum of the series to n terms whose general term is 2n + 1. Solution : Sn = T n = (2n + 1) = 2n + 1 =
2(n 1) n + n = n2 + 2n 2 n
Example # 36 : T k = k 2 + 2k , then find n
Solution :
k 1
=
k
.
k 1
n
n
Tk =
T
k2 +
2
k
k 1
k 1
n (n 1) ( 2n 1) n (n 1) ( 2n 1) 2(2n 1) + = + 2n + 1 – 2. 6 6 2 1
n
Example # 37 : Find the value of the expression
i
j
1
i 1 j 1k 1
n
Solution :
i
n
j
1
=
i 1 j 1k 1
i
i 1 j 1
j
n i (i 1) 1 i2 = = 2 2 i 1 i 1 n
RESONANCE
n
i1
i
=
1 n (n 1) ( 2n 1) n (n 1) 6 2 2
=
n (n 1) (n 2) n (n 1) [2n + 1 + 3] = . 6 12
14
Sequence & Series : PAJ Sir
Method of difference for finding n th term : Let u1, u2, u3 ........ be a sequence, such that u2 – u1, u3 – u2, ......... is either an A.P. or a G.P. then nth term un of this sequence is obtained as follows S = u1 + u2 + u3 + ........... + un ................(i) S= u1 + u2 + ........... + un–1 + un ................(ii) (i) – (ii) un = u1 + (u2 – u1) + (u3 – u2) + ........... + (un – un–1) Where the series (u2 – u1) + (u3 – u2) + .......... + (un – un–1) is either in A.P. or in G.P. then we can find un. n
So sum of series S =
u
r
r 1
Note : The above method can be generalised as follows : Let u1, u2, u3, ......... be a given sequence. The first differences are 1u1, 1u2, 1u3, ........ where 1u1 = u2 – u1, 1u2 = u3 – u2 etc. The second differences are 2u1, 2u2, 2u3, ......., where 2u1 = 1u2 – 1u1, 2u2 = 1u3 – 1u2 etc. This process is continued untill the kth differences ku1 , ku2 , ........ are obtained, where the kth differences are all equal or they form a GP with common ratio different from 1. Case - 1 : The kth differences are all equal. In this case the nth term, un is given by k k–1 un = a0n + a1n + .....+ ak , where a0, a1, ...., ak are calculated by using first 'k + 1' terms of the sequence. Case - 2 : The kth differences are in GP with common ratio r (r 1) The nth term is given by un = rn – 1 + a0 nk–1 + a1 nk–2 + ..... + ak–1 Example # 38 : Find the sum to n-terms 3 + 7 + 13 + 21 + ......... Solution :
Let
S = 3 + 7 + 13 + 21 + ......... + Tn ...........(i) S= 3 + 7 + 13 + ............ + Tn–1 + Tn ...........(ii) (i) – (ii) Tn = 3 + 4 + 6 + 8 + .......... + (Tn – Tn–1)
n 1 [8 + (n – 2)2] 2 = 3 + (n – 1) (n + 2) = n2 + n + 1 Hence S = (n2 + n + 1) = n2 + n + 1 =3+
=
n(n 1)(2n 1) n n(n 1) + +n = (n2 + 3n + 5) 6 3 2
Example # 39 : Find the sum to n-terms 1 + 4 + 10 + 22 + ........ Solution :
Let
S = 1 + 4 + 10 + 22 + .........+ Tn ........(i) S= 1 + 4 + 10 + ......... + Tn–1 + Tn ........(ii) (i) – (ii) Tn = 1 + (3 + 6 + 12 + ......... + Tn – Tn–1 )
So
RESONANCE
2n1 1 Tn = 1 + 3 2 1 Tn = 3 . 2n–1 – 2 S = Tn = 3 2n–1 – 2
15
Sequence & Series : PAJ Sir
2n 1 = 3 . 2 1 – 2n = 3.2n – 2n – 3
Method of difference for finding s n : If possible express rth term as difference of two terms as tr = ± (f(r) – f(r ± 1)). This can be explained with the help of examples given below. t1 = f(1) – f(0), t2 = f(2) – f(1),
tn = f(n) – f(n-1) Sn = f(n) – f(0)
Example # 40 : Find the sum of n-terms of the series 1.2 + 2.3 + 3.4 + ............ Solution :
Let Tr be the general term of the series So Tr = r(r + 1). To express tr = f(r) – f(r–1) multiply and divide tr by [(r + 2) – (r – 1)] so
Tr
Let f(r) = so
=
r (r + 1) [(r + 2) – (r – 1)] 3
=
1 [r (r + 1) (r + 2) – (r – 1) r (r + 1)]. 3
1 r (r + 1) (r + 2) 3
Tr = [f(r) – f(r – 1)]. n
Now S =
T
r
r 1
= T1 + T2 + T3 + .........+ Tn
T1 =
1 [1 . 2 . 3 – 0] 3
T2 =
1 [2 . 3 . 4 – 1 . 2 . 3] 3
T3 =
1 [3 . 4 . 5 – 2 . 3 . 4] 3
Tn =
1 [n(n+1) (n + 2) – (n – 1)n (n + 1)] 3
S=
1 n (n + 1) (n + 2) 3
Hence sum of series is f(n) – f(0).
Example # 41 : Sum to n terms of the series Solution :
1 1 1 + + + ......... (1 x )(1 2x ) (1 2x )(1 3 x ) (1 3 x )(1 4x )
Let Tr be the general term of the series
RESONANCE
16
Sequence & Series : PAJ Sir Tr =
1 (1 rx )(1 (r 1)x )
So
Tr =
1 1 (1 (r 1)x ) (1 rx ) 1 1 = ( 1 rx )( 1 ( r 1 ) x ) 1 rx 1 ( r 1 ) x x x
Tr = f(r) – f(r + 1) S = Tr = T1 + T2 + T3 + .......... + Tn =
=
1 1 1 x 1 x 1 (n 1)x
n (1 x )[1 (n 1)x]
4 5 6 Example # 42 : Sum to n terms of the series 1 . 2 . 3 + 2 . 3 . 4 + 3 . 4 . 5 + ......... Solution :
Let Tr
=
r 3 r(r 1)(r 2)
=
1 3 + (r 1)(r 2) r(r 1)(r 2)
1 1 1 3 1 = + r 1 r 2 2 r(r 1) (r 1)(r 2)
1 1 1 3 1 S= + 2 n 2 2 ( n 1 )( n 2 ) 2
=
3 1 5 1 5 – – [2n + 5] 1 = 2 ( n 1 ) 2(n 1)(n 2) 4 n2 4
Example # 43 : Find the nth term and the sum of n term of the series 2 + 12 + 36 + 80 + 150 + 252 + .......... Solution :
Let
S = 2 + 12 + 36 + 80 + 150 + 252 + ................+Tn S= 2 + 12 + 36 + 80 + 150 + 252 + .........+Tn–1 + Tn (i) – (ii) Tn = 2 + 10 + 24 + 44 + 70 + 102 + ............... + (Tn – Tn–1) Tn = 2 + 10 + 24 + 44 + 70 + 102 + ....... + (Tn–1–Tn–2) + (Tn – Tn–1) (iii) – (iv) Tn – Tn–1 = 2 + 8 + 14 + 20 + 26 + .........
n [4 + (n – 1) 6] = n [3n – 1] Tn – Tn–1 = 3n2 – n 2 general term of given series is (Tn – Tn–1) = (3n2 – n) = n3 + n2. Hence sum of this series is S = n3 + n2 =
=
n(n 1)(2n 1) n2 (n 1)2 + 6 4
=
n (n 1) (3n2 + 7n + 2) 12
=
RESONANCE
1 n (n + 1) (n + 2) (3n + 1) 12
17
...........(i) ...........(ii) ...........(iii) ...........(iv)
Sequence & Series : PAJ Sir Example # 44 : Find the general term and sum of n terms of the series 9 + 16 + 29 + 54 + 103 + ......... Solution :
Let
S = 9 + 16 + 29 + 54 + 103 + ................. + Tn S= 9 + 16 + 29 + 54 + 103 + ......... + Tn–1 +Tn (i) – (ii) Tn = 9 + 7 + 13 + 25 + 49 + ................. + (Tn – Tn–1) Tn = 9 + 7 + 13 + 25 + 49 + ........... + (Tn–1–Tn–2) + (Tn – Tn–1)
...........(i) ...........(ii) ...........(iii) ...........(iv)
n–2 n–2 12 24 ........ (iii) – (iv) Tn – Tn–1 = 9 + (–2) + 6 = 7 + 6 [2 – 1] = 6(2) + 1.
(n 2 ) terms
General term is Tn = 6(2) + n + 2 Also sum S = Tn = 62n–1 + n + 2 n–1
=6.
n (n 1) (2n 1) + + 2n 2 2 1
= 6(2n – 1) +
n(n 5 ) 2
Self practice problems : (25)
Sum to n terms the following series (i)
1 3
1
+
1 2 1 2 3
3
+
1 2 3 1 23 3 3 3
+ .........
(ii)
1 1 1 + + 1. 3 . 5 3.5.7 5 . 7 . 9 + .........
(iii)
1 . 5 . 9 + 2 . 6 . 10 + 3 . 7. 11 + .........
(iv)
4 + 14 + 30 + 52 + 80 + 114 + ..........
(v)
2 + 5 + 12 + 31 + 86 + ...............
Answers :
(25)
(i)
(iii) (iv) (v)
RESONANCE
2n n 1
(ii)
1 1 1 4 3 (2n 1)(2n 3)
n (n + 1) (n + 8) (n + 9) 4 n(n + 1)2
3n n2 n 1 2
18
