Find $r$ of G.P with arithmetic progression members

Question

$a1$$ \quad $$a3$$ \quad $$a13$$ \quad$ of arithmetic progression make G.P, find $r$ of G.P

I tried this: $$ a1 = a \qquad a3 = a + 2d \qquad a13 = a + 12d $$ $$ r = \frac{a+2d}{a} $$ then putting this into $Sn$ but it's not working out.

Answer 1

Because you didn't do it completely.

Since you have $$r =\frac{a + 2d}{a} \quad(a \neq 0) $$ then you get:

$$\frac{a + 2d}{a} * (a + 2d) = a + 12d. $$

By simple simplification, you'll get $$d = 2a$$.

Then you get $$a1 = a \quad a3 = 5a \quad a13 = 25a $$ as your desire. Where $$r = 5 $$