Question

Show that in an infinite G.P., each term bears a constant ratio to the sum of all
the terms that follow it.
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Answer 1

Proof that in an infinite G.P., each term bears a constant ratio to the sum of all the terms that follow it.

Step-by-step explanation:

If the first term of the GP is a and common ratio r

The n th term is given by

$t_{n}=ar^{n-1}$

The terms that follow it are

$ar^n, ar^{n+1}, ar^{n+2},....$

Sum of the above terms till infinity is

$S=\frac{ar^n}{1-r}$

The ratio of nth term and sum of the the terms that follow it

$\frac{t_n}{S}=ar^{n-1}\times \frac{1-r}{ar^n}$

$\implies \frac{t_n}{S}=\frac{1-r}{r}$

Since the common ratio r is constant therefore the RHS is constant.

Hope this helps.

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