ACTIVITY 12: MORE PROBLEMS IN GEOMETRIC SEQUENCES Do the followi!: 1" C#e$te C#e$te $ %o%e&t %o%e&t we' we' fo# !eo( !eo(et# et#i% i% )e*+e )e*+e%e) %e)""
GEOMETRIC SEQUENCE
A geometric sequence is a sequence where each term after the rst term is otaine! " mu#ti$#"ing the $rece!ing term " a non%ero constant ca##e! the common ratio&
'in!ing the nth term of Geometric sequence
The nth term of of a geometric sequence is a n( a)rn*) where a) is the rst term. an is the nth term. an! r is the common ratio&
Sum of nite an! innite geometric sequence
Geometric Means
Geometric means are the terms etween an" two non*successi/e terms in a geometric sequence& The geometric mean etween two numers a an! is 2a if a an! are $ositi/e. *2a if a
The formu#a formu#a for the sum of the the rst n terms in a geometric sequence is Sn( a)*a)rn or Sn( a)+)*rn, )*r
)*r
-here Sn is the sum. a ) is the rst term an! r is the common ratio if r is not equa# to )& The sum of the terms of an innite geometric sequence is S( a) !i/i!e! " )*r. where a ) is the rst term an! r is the common ratio an! I r I 0 )& If r( ). Sn(na )& If r( *) an! n is e/en. Sn(1 If r( *) an! n is o!!. Sn( a )
2" Co(&$#e Co(&$#e $, %ot#$)t %ot#$)t $#ith(et $#ith(eti% i% $, !eo(et#i !eo(et#i%% )e*+e%e) )e*+e%e) +)i! $ two-%ol two-%ol+( +( %h$#t" %h$#t"
A#it A#ith( h(et eti% i% Se*+ Se*+e e%%e
Geo( Geo(et et#i #i%% Se*+ Se*+e e%e %e
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The th te#( i) ,efie, $)
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A %o)t$t i) 'ei! $,,e,
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7$) $ %o((o #$tio The th te#( i) ,efie, $) an( a)rn*) A %o)t$t i) 'ei! (+lti&lie,
." Gi/e Gi/e the !eo(et#i !eo(et#i%% )e*+e%e )e*+e%e 10 20 0 0 130 .20 40 thi5 thi5 of $ )i(&le )i(&le #e$l-li #e$l-life fe )it+$ti )it+$tio o whi%h #e&#e)et) thi) )e*+e%e" 6+)ti h$) $ #e%t$!+l$# &ie%e of &$&e#" A) he fol,) the &$&e# ito h$lf0 two #e%t$!le) we#e ($,e" 7e t#ie, to fol, it $!$i $, h$, fo+# #e%t$!le)" A!$i he fol,e, it $, h$, ei!ht #e%t$!le)" 7e #e&e$te, the &#o%e,+#e $, e,e, +& (+lti&l8i! the l$)t te#( '8 two to fi, the e9t te#("
Se*+e%e: 10 20 0 0 130 .2 No of fol,)
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an( a)rn*) 9=11>;9=1 ? ;9=1>9=2 @;9=1@;9=1 ? @9=11@9=2 2;92=19=1 ? 92 =1.9=22 2;92-92=19-1.9=1-22 ? 292-.9-21 ? O# 92-9- ?
9=?
9-1?
9? -
9? 1
? -> S? $1@1-# >1-#
S1? .@1-21>1-2 S1?.@1-12>-1 S1?.@-12.>-1 S1? -.3> -1 S1? .03
The sum of the rst )1 terms is 7189&
;" 9-2 ? 9-2>29 @9-2@9-2 ? 29@19-2 92-29=? .92-9 92-.92-29=9=? 1192-29=?
The /a#ues of 3 are are :5)) an! :&
119-2? 9-2? 119?2 9?2 ? 2>11 3" The Fo# Fo#l, l, 7e$lth O#!$i$ O#!$i$tio tio #e&o#te, #e&o#te, th$t $'o+t 13 (illio (illio $,ole)%et $,ole)%et !i#l) !i#l) 'etwee 'etwee 1; $, 1 8e$#) of $!e !i/e 'i#th e$%h 8e$#" Howi! the $,/e#)e effe%t) of $,ole)%et %hil,'e$#i! o the he$lth of the (othe#) $) well $) thei# if$t)0 $ !#o+& of )t+,et) f#o( M$!iti! 7i!h S%hool /ol+tee#e, to hel& the !o/e#(et i it) %$(&$i! fo# the &#e/etio of e$#l8 &#e!$%8 '8 !i/i! le%t+#e) to '$#$!$8) $'o+t the F7O G+i,elie) o tee$!e &#e!$%8" Gi#l) f#o( othe# '$#$!$8 he$#, $'o+t it0 )o !i#l) $tte,e, f#o( B$#$!$8 20 13 fo# B$#$!$8 .0 $, )o o" $" M$5e M$5e $ t$'l t$'lee #e&# #e&#e) e)e eti ti! ! the the +(' +('e# e# of $,ol $,ole) e)%e %ett !i#l !i#l)) who who $tte $tte, ,e, e, the the le%t le%t+# +#ee f#o( f#o( B$#$!$8 1 to B$#$!$8 $))+(i! th$t the +('e#) of $tte,ee) ,o+'le) ,o+' le) $t e$%h '$#$!$8" B$#$!$8
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No" of !i#l) who $tte,e,
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'" A$l8e A$l8e the ,$t$ i the t$'le $, %#e$te $ fo#(+l$" fo#(+l$" U)e the fo#(+l$ fo#(+l$ to +)tif8 +)tif8 8o+# ,$t$ i the t$'le" 1-2 ?13. The tot$l +('e# of !i#l) who 'eifite, f#o( the le%t+#e i) 130 ."
MA; T
1" I 8o+# 8o+# #e)& #e)&e%t e%ti/ i/ee !#o+& !#o+&00 ($5e ($5e +)e +)e of $ll $ll the the %o%e %o%e&t &t)) 8o+ 8o+ le$# le$#e, e, o !eo( !eo(et et#i #i%% )e*+ )e*+e e%e) %e) %o)i,e#i! the )t$#ti! )$l$#80 the #$te of i%#e$)e0 ti(e f#$(e0 et%" i ($5i! ,iffe#et )$l$#8 )%he(e) $, i ,e%i,i! whi%h oe will 'e the 'e)t fo# 'oth the e(&lo8e# $, the wo+l,-'e e(&lo8ee)" 2" P#e&$# P#e&$#ee $ /i)+$l /i)+$l &#e)et &#e)et$ti $tio o of 8o+# %ho)e %ho)e )$l$#8 )$l$#8 )%he(e )%he(e with the ,iffe#e ,iffe#et t ,$t$ th$t we#e +)e,0 +)e,0 to!ethe# with the fo#(+l$) $, $ll the %o(&+t$tio) ,oe" Yo+ ($8 i%l+,e oe o# two )$l$#8 )%he(e) th$t 8o+ h$/e &#e&$#e, i 8o+# !#o+& fo# %o(&$#i)o)" ." I $ )i(+l$te, )i(+l$te, 'o$#, (eeti!0 (eeti!0 )how )how 8o+# /i)+$l /i)+$l &#e)et$ti &#e)et$tio o to 8o+# %l$))($te %l$))($te)) who will $%t $) the the h+($ #e)o+#%e $,(ii)t#$ti/e offi%e#)"
Ae)h$ w$) $)5e, to %#e$te $ )$l$#8 )%he(e fo# $ i(&o#t$t o' th$t thei# %o(&$8 wo+l, offe# to the 'e)t IT !#$,+$te)" She offe#e, 10 Ph& $) )t$#ti! )$l$#8 $, will #e%ei/e 2J $,,itio$l o the l$)t )$l$#8 e/e#8 fo+# (oth)" Fh$t will 'e the )$l$#8 o the 2th (othK 7ow (+%h will 'e hi) tot$l e$#i!) i 1 8e$#)K 6$+$#8-A&#il
M$8-A+!+)t
1 Ph&
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Se&te('e#De%e('e# 1 Ph&
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M$8-A+!+)t
1 2 Ph&
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Se&te('e#De%e('e# 2 ."2 Ph&
$? $1# -1 2>? 3 $? @1 @1"2 ; $? @1@2".2 $? 2 ." 2 The wo+l,-'e e(&lo8ee) will #e%ei/e 2 ."2 Ph& o the 2th (oth"
S? $1@1-# >1-# S3? @1@1-2";>1-1"2 S3? @1@-1";>-"2 )3? -1;">-"2 )3?2"2 The wo+l, 'e e(&lo8ee) will h$/e 2"2 Ph& $) tot$l e$#i! $t the e, of the 2, 8e$#"
