If a, b, c be the pth, qth, rth terms of a GP, then the value of (q �

1.	If a, b, c be the pth, qth, rth terms of a GP, then the value of (q – r) log a + (r – p) log b + (p – q) log c is : (a) 0 (b) 1 (c) –1 (d) pqr
Answer» (a) 0 Let h be the first term and k be the common ratio of a GP, then a = hk^{p – 1}, b = hk^{q – 1}, c = hk^{r – 1} ∴ (q – r) log a + (r – p) log b + (p – q) log c = log [hk^{p –1}]^{q – r} + log [hk^{q –1}]^{r – p} + log[hk^{r –1}]^{p – q} = log(h^{q – r + r – p + p – q}) (k^{p – 1})^{q – r} (k^{q –1})^{r – p} (k^{r –1})^{p – q} = log(h^o k^o) = log 1 = 0.

If a, b, c be the pth, qth, rth terms of a GP, then the value of (q – r) log a + (r – p) log b + (p – q) log c is : (a) 0 (b) 1 (c) –1 (d) pqr

Answer»

(a) 0

Let h be the first term and k be the common ratio of a GP, then

a = hk^{p – 1}, b = hk^{q – 1}, c = hk^{r – 1}

∴ (q – r) log a + (r – p) log b + (p – q) log c

= log [hk^{p –1}]^{q – r} + log [hk^{q –1}]^{r – p} + log[hk^{r –1}]^{p – q}

= log(h^{q – r + r – p + p – q}) (k^{p – 1})^{q – r} (k^{q –1})^{r – p} (k^{r –1})^{p – q}

= log(h^o k^o) = log 1 = 0.