IPMAT Indore 2024 - QA (MCQ) Q38 Explanation

Explanation:

(d) We know nth term of a G.P. is given by the formula,

a n = ar^n-1 

Using above formula, we can say,

pth term = ar^p – 1 = q …..(1)

qth term = ar^q – 1 = p …..(2)

where, a is the first term and r is the common ratio of the G.P.

Diving equation (2) by equation (1), we get

p/q = r^q−1 / r^p−1

p/q = r^q−p

Taking log both sides, we get log (p/q) = (q−p)logr …..(3)

Taking log on both sides, of equation (2), we get

log p = log a + log r^q−1

log p = log a + (q – 1) log r

log a = log p – (q – 1) log r ….(4)

Substituting the value of ‘log r’ from eqn.(3) in eqn.(4), we get

log a = log p – (q – 1). log (p/q) . 1/(q−p)

log a = log p + (q−1)[logp − logq]/(p−q)

log a = (p – q) log p + (q – 1) log p – (q – 1) log q

log a = (p−1) logp + (1−q) logq/p−q

or, loga= (1−q) logq − (1−p) logp/p−q Option (d) Ans.