Question

If the pth, qth and rth terms of a G.P.are in geometric progression. How to show,that p, q, and r are successive terms of an Arithmetic progression?

Answer 1

The Proof is given in the Explanation.

Explanation:

Let, the Geom. Prog., be, `a, ab, ab^2,..., ab^(n-1),...; n in NN, a!=0.`

Its `n^(th)" term, "t_n," is "ab^(n-1),.............(ast) (n in NN).`

We are given that, `t_p, t_q, and, t_r` are in GP, i.e.,

`t_q/t_p=t_r/t_q,..........."[=Common Ratio],"`

`rArr (t_q)^2=t_p*t_r..................(star).`

Using `(ast)` to determine `t_p,t_q,t_r,` we find,

`t_p=ab^(p-1), t_q=ab^(q-1), and, t_s=ab^(r-1).`

Then, by `(star),` we have,

`{ab^(q-1)}^2={ab^(p-1)*ab^(r-1)}, or,`

`a^2b^(2(q-1))=a^2b^((p-1)+(r-1))=a^2b^(p+r-2).`

Cancelling out `a^2!=0` from both sides, we get,

`b^(2(q-1))=b^(p+r-2), i.e., b^(2q-2)=b^(p+r-2).`

`rArr 2q-2=p+r-2, or, 2q=p+r rArr q-p=r-q`

`rArr p, q," and, "r" are in A.P."`

Hence, the Proof.