Question

If a, b, c are p^th, q^th and r^th terms respectively of a G.P., then prove that

`|(log a, p, 1),(log b, q, 1),(log c, r, 1)| = 0`

Answer 1

Suppose A is the first term and R is the common ratio of the G.P.

Then, a = AR^{p – 1}   ...(i)

b = AR^{q – 1}   ...(ii)

c = AR^{r – 1}   ...(iii)

From (i), (ii) and (iii)

log a = log A + (p – 1) log R

log b = log A + (q – 1) log R

log c = log A + (r – 1) log R

∴ `|(loga, p, 1),(logb, q, 1),(logc, r, 1)| = |(log A + (p - 1) log R, p, 1),(log A + (q - 1) log R, q, 1),(log A + (r - 1) log R, r, 1)|`

= `|(log A + (p - 1) log R, p - 1, 1),(log A + (q - 1) log R, q - 1, 1),(log A + (r - 1) log R, r - 1, 1)|`   ...[Applying C₂ → C₂ – C₃]

= `|(0, p - 1, 1),(0, q - 1, 1),(0, r - 1, 1)|`   ...[Applying C₁ → C₁ – logA C₃ – logR C₂]

⇒ `|(loga, p, 1),(logb, q, 1),(logc, r, 1)| = 0`

Hence Proved.