Find all sequences which are simultaneously A.P. and G.P.

Explanation

Let a1,a2,a3,...,an,... be a sequence which is both an AP as well as a GP.

Consider three consecutive terms` a_n,a_(n+1),a_(n+2)` of this AP and GP.

Then,` 2a_(n+1)=a_n+a_(n+2) ...(i)`

And, if r be the common ratio of the GP, then


`a_n=a_1r^(n−1),a_(n+1)=a_1r^n` and `a_(n+2)=a_1r^(n+1)`

Putting these values in (i), we get

`2a_1r^n=a_1r^(n−1)=>r^2−2r+1=0`

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