Question

If a, b, c are positive and are the p^th, q^th and r^th terms respectively of a G.P., then the value of `|(loga, p, 1),(logb, q, 1),(logc, r, 1)|` is ______.

Options

• 0

• p

• q

• r

Answer 1

If a, b, c are positive and are the p^th, q^th and r^th terms respectively of a G.P., then the value of `|(loga, p, 1),(logb, q, 1),(logc, r, 1)|` is 0.

Explanation:

Let a be the first term and R be the common ratio of the G.P., then

a = AR^p–1

⇒ loga = logA + (p – 1) logR  ...(i)

b = AR^q–1

⇒ logb = logA + (q – 1) logR  ...(ii)

c = AR^r–1 

⇒ logc = logA + (r – 1) logR  ...(iii)

Now, multiplying (i), (ii) and (iii) by (q – r), (r – p) and (p – q) respectively and adding, we get

loga (q – r) + logb (r – p) + logc (p – q) = 0

⇒ Δ = 0