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The sum of the first ten terms of an A.P. is 160 and the sum of the first two terms of a G.P. is 8. If the first term of the A.P. is equal to the common ratio of the G.P. and the first term of the G.P. is equal to common difference of the A.P., then the sum of all possible values of the first term of the G.P. is:
Sum of first $$n$$ terms of an AP: $$S_n = \frac{n}{2}[2a + (n-1)d]$$
Sum of first $$2$$ terms of a GP: $$S_2 = A + Ar$$
$$S_{10} = \frac{10}{2}[2a + 9d] = 160$$
$$5[2r + 9A] = 160$$
$$2r + 9A = 32 \implies r = \frac{32 - 9A}{2}$$
$$A + Ar = 8 \implies A(1 + r) = 8$$
$$A\left(1 + \frac{32 - 9A}{2}\right) = 8$$
$$9A^2 - 34A + 16 = 0$$
$$\text{Sum of values of } A = \frac{34}{9}$$
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