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  • In a geometric progression consisting of positive terms, each term equals the sum of the next two terms. Then the common ratio of this progression equalsOpt

In a geometric progression consisting of positive terms, each term equals the sum of the next two terms. Then the common ratio of this progression equals

Option: 1

\sqrt{5}\; \;


Option: 2

\; \frac{1}{2}\left ( \sqrt{5}-1 \right )\; \;


Option: 3

\; \frac{1}{2}\left ( 1-\sqrt{5} \right )\;


Option: 4

\; \; \frac{1}{2}\sqrt{5}


Answers (1)

best_answer

Use the concept of

General term of a GP

T_{n}= ar^{n-1}
where

a\rightarrow first term

r\rightarrow common ratio

 

Now,

Let the G.P be a, ar, ar2  ...

Given that Tn =  Tn+1  + Tn+2

 So that arn-1 =arn +arn+1

\therefore \tfrac{r^{n}}{r}=r^{n} + r^{n}.r

\therefore \frac{1}{r}= 1 + r

r^{2} + r-1= 0

r =\frac{ \sqrt{5}-1}{2}

(Negative value rejected as all terms of GP are positive)

Posted by

mansi

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