Geometric progression

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If the first term of an infinite G.P. is 1 and each term is twice the sum of the succeeding terms, then the common ratio is

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Solution

Let the G.P. be 1 $r, r^{2}, r^{3}, . .$
Given, $t_{n} = 2 (t_{n + 1} + t_{n + 2} + t_{n + 3} + \dots to \infty)$
$\begin{array}{l} or t_{n} = 2 \cdot \frac{t_{n + 1}}{1 - r}, (∵ common ratio = r) \\ ∴ = \frac{t_{n + 1}}{t_{n}}; or \frac{1 - r}{2} = r \\ or 1 - r = 2 r; ∴ r = \frac{1}{3} \end{array}$

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Geometric progression

Remember concepts with our Masterclasses.

If the first term of an infinite G.P. is 1 and each term is twice the sum of the succeeding terms, then the common ratio is

Let the G.P. be 1 r,r2,r3,..Given, tn=2tn+1+tn+2+tn+3+… to ∞ or tn=2⋅tn+11−r, (∵ common ratio =r)∴ =tn+1tn; or 1−r2=r or 1−r=2r;∴r=13

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