Question

If Pth
, q
th and rth terms of an A.P. are in G.P. then the common ratio of this G.P. is?​

Answer 1

Answer:

Let first term of the AP = a and common difference = d

pth of an AP= a+(p−1)d

qth of an AP= a+(q−1)d

rth of an AP= a+(r−1)d

are in G.P

Let first term of the GP be x and common ratio  be y

then

a+(p−1)d=x........(i)

a+(q−1)d =xy........(ii)

a+(r−1)d=xy² ........(iii)

Subtract (i) from (ii) and get

(q−1)d −(p−1)d=xy−x

→(q−p)d=xy−x......(iv)

Subtract (ii) from (iii) and get

(p−1)d −(r−1)d=xy²−xy =y(xy−x)

→(q−r)d=y(xy−x)  .....(v)

divide (v) by (iv) and get

∵ y = q-r/p-q