Name: Sum of Infinite Terms of G.P.
Uploaded: 2019-07-19
Duration: 3 min 36 s
Description: Do you know that we can find the sum of an infinite number of terms of G.P.? Of course, there will be some conditions. Check this video to learn more.

Question

If the

p^{th}, q^{th} & r^{th}

terms of an A.P. are in G.P., then find the common ratio of the G.P.

Accepted Answer

Let $a$ be the first term and $d$ be the common difference of the given A.P.

Then, we have:

$p^{th}$ term, $a_{p} = a + (p - 1) d$

$q^{th}$ term, $a_{q} = a + (q - 1) d$

$r^{th}$ term, $a_{r} = a + (r - 1) d$

Now, according to the question the $p^{th}$ , the $q^{th}$ and the $r^{th}$ terms are in G.P.

$∴ (a + (q - 1) d)^{2} = (a + (p - 1) d) \times (a + (r - 1) d)$

$\Rightarrow a^{2} + 2 a (q - 1) d + ((q - 1) d)^{2} = a^{2} + a d (r - 1 + p - 1) + (p - 1) (r - 1) d^{2}$

$\Rightarrow a d (2 q - 2 - r - p + 2) + d^{2} (q^{2} - 2 q + 1 - p r + p + r - 1) = 0$

$\Rightarrow a (2 q - r - p) + d (q^{2} - 2 q - p r + p + r) = 0$ ( $∵ d$ cannot be $0$ )

$\Rightarrow a = - \frac{(q^{2} - 2 q - p r + p + r) d}{(2 q - r - p)}$

$∴$ Common ratio $= \frac{a_{q}}{a_{p}}$

$= \frac{a + (q - 1) d}{a + (p - 1) d}$

$= \frac{\frac{(q^{2} - 2 q - m + p + r) d}{(p + r - 2 q)} + (q - 1) d}{\frac{(q^{2} - 2 q - m + p + r) d}{(p + r - 2 q)} + (p - 1) d}$

$= \frac{q^{2} - 2 q - p r + p + r + p q + r q - 2 q^{2} - p - r + 2 q}{q^{2} - 2 q - p r + p + r + p^{2} + p r - 2 p q - p - r + 2 q}$

$= \frac{p q - p r - q^{2} + q r}{p^{2} + q^{2} - 2 p q}$

$= \frac{p (q - r) - q (q - r)}{(p - q)^{2}}$

$= \frac{(p - q) (q - r)}{(p - q)^{2}}$

$= \frac{(q - r)}{(p - q)}$

If the $p^{th}, q^{th} & r^{th}$ terms of an A.P. are in G.P., then find the common ratio of the G.P.

Important Questions on Sequences and Series

Learn and Prepare for any exam you want !

If the pth, qth & rth terms of an A.P. are in G.P., then find the common ratio of the G.P.

Important Questions on Sequences and Series

Learn and Prepare for any exam you want !

If the $p^{th}, q^{th} & r^{th}$ terms of an A.P. are in G.P., then find the common ratio of the G.P.