# An arithmetic & geometric progression question.

#### misogynisticfeminist

I have an arithmetic series, with the sum of the first n terms to be 610. The 1st, 3rd and 11th terms of this AP is the same as the 3rd, 2nd and 1st term of a geometric series. Find the first term of the geometric series.

I have constructed 4 equations from this

$$a_p = a_q r^2$$

$$a_p+2d = a_q r$$

$$a_p + 10d =a_q$$

$$20a_p +19d =610$$

where p represents the AP and Q represents the GP.

But i seem to have problems solving them simultaneously. Can anyone provide some insight?

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