Question

if the pth, qth, rth term of an ap are in gp, then find the common ratio of the go
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Answer 1

Answer:

$r=\frac{q-r}{p-q}$

Step-by-step explanation:

In AP, first term be a

Difference be d

∴p  th  term ⇒a+(p−1)d

q  th  term a+(q−1)d

r  th  term a+(r−1)d

since they are in GP

and let first term in G.P be A

and common ration be r

∴ As given

a+(p−1)d=A...(1)

a+(q−1)d=Ar...(2)

a+(r−1d)=$AR^{2}$ ...(3)

subtracting (1) from (2)

[(q−1)−(p−1)]d=A[r−1]

(q−p)d=A(r−1)...(4)

subtracting (2) form (3)

(r−q)d=Ar(r−1)...(5)

Dividing (5), we get

$\frac{Ar(r-1)}{A(r-1)} =\frac{(r-q)d}{(q-p)d}$

​  ∴$r=\frac{q-r}{p-q}$

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