Two positive numbers have arithmetic mean 25 and geometric mean 15…

(a), (b) $a + b = 25 \times 2 = 50$ and $ab = 15^2 = 225$. So $a$ and $b$ are the roots of $t^2 - 50t + 225 = 0$, that is $(t - 5)(t - 45) = 0$: the numbers are $45$ and $5$.

(c) Six terms from $5$ to $45$: $d = \dfrac{45 - 5}{5} = 8$, so the means are $13, 21, 29, 37$. Their sum is $100$, which is $4 \times 25$, as expected: each pair from the two ends adds up to $5 + 45$.

(d) Five terms from $5$ to $45$: $r^4 = 9$, so $r = \sqrt{3}$ and the means are $5 \sqrt{3}, 15, 15 \sqrt{3}$. Their product is $3375 = 15^3 = G^3$, since the middle mean is $G$ and the outer two multiply to $G^2$.

(e) The AP is $5, 13, 21, \dots$ with $d = 8$: $S_n = \dfrac n2\,[10 + 8(n - 1)] = n(4n + 1)$. Now $S_{16-1} = 915$ and $S_{16} = 1040$, so the least number of terms is $16$.