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If-the-sum-of-an-infinite-GP-be-3-and-the-sum-of-the-squares-of-its-term-is-also-3-then-its-first-term-and-common-ratio-are-




Question Number 43629 by peter frank last updated on 12/Sep/18
If  the sum of an infinite GP be 3 and the  sum of the squares of its term is also 3,  then its first term and common ratio are
IfthesumofaninfiniteGPbe3andthesumofthesquaresofitstermisalso3,thenitsfirsttermandcommonratioare
Answered by $@ty@m last updated on 13/Sep/18
ATQ,  a+ar+ar^2 +.....=3  ⇒(a/(1−r))=3 ...(1)  Again,  (a)^2 +(ar)^2 +(ar^2 )^2 +(ar^3 )^2 +......=3  ⇒a^2 (1+r^2 +r^4 +r^6 +....)=3  ⇒a^2 ×(1/(1−r^2 ))=3 ...(2)  from (1) &(2),  (a/(1−r))=(a^2 /(1−r^2 ))  ⇒a=1+r ...(3)  from (1) &(3)  ((1+r)/(1−r))=3  ⇒r=(1/2) Ans.  and from (3)  a=(3/2) Ans.
ATQ,a+ar+ar2+..=3a1r=3(1)Again,(a)2+(ar)2+(ar2)2+(ar3)2+=3a2(1+r2+r4+r6+.)=3a2×11r2=3(2)from(1)&(2),a1r=a21r2a=1+r(3)from(1)&(3)1+r1r=3r=12Ans.andfrom(3)a=32Ans.

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