If pth, qth, and rth terms of a G.P. are x, y, z respectively, then

1.	If pth, qth, and rth terms of a G.P. are x, y, z respectively, then write the value of xq-r yr-p, zp-q.
Answer» Let the first term be a and the common ratio be R. ∴ According to the question, a_p = x. a_q = t a_r= z. We know that a_n = aR^n-1 ∴ a_p = aR^p-1= x a_q = aR^q-1= y a_r = aR^r-1= z ⇒ x^q-r = (aR^p-1)^q-r ⇒ y ^r-p= (aR^q-1)^r-p ⇒ z^p-q = (aR^r-1)^p-q Multiplying the above three equations we get x ^q-r .y ^r-p.z ^p-q = (a^q-r .R^pq-pr-q+r). (a^r-p.R^rq-pq-r+p). (a^p-q.R^pr-qr-p+q) =(a ^q-r+r-p+p-q.R ^{pq-pr-q+r+rq-pq-r+p+pr-qr-p+q}) = (a⁰.R⁰) = 1.

If pth, qth, and rth terms of a G.P. are x, y, z respectively, then write the value of xq-r yr-p, zp-q.

Answer»

Let the first term be a and the common ratio be R.

∴ According to the question,

a_p = x.

a_q = t

a_r= z.

We know that a_n = aR^n-1

∴ a_p = aR^p-1= x

a_q = aR^q-1= y

a_r = aR^r-1= z

⇒ x^q-r = (aR^p-1)^q-r

⇒ y ^r-p= (aR^q-1)^r-p

⇒ z^p-q = (aR^r-1)^p-q

Multiplying the above three equations we get

x ^q-r .y ^r-p.z ^p-q

= (a^q-r .R^pq-pr-q+r). (a^r-p.R^rq-pq-r+p). (a^p-q.R^pr-qr-p+q)

=(a ^q-r+r-p+p-q.R ^{pq-pr-q+r+rq-pq-r+p+pr-qr-p+q}) = (a⁰.R⁰)

= 1.