+280 Let \(a_1, a_2, a_3, \dots\) be an arithmetic sequence. Let \(S_n\) denote the sum of the first \(n\) terms. If \(S_5 = \frac{1}{5}\) and \(S_{10} = \frac{1}{10},\) then find \(S_{15}.\)
+129846 Sum of an arithmetic series from a1 to an inclusive = n * a1 + (n)(n-1) / 2 * d
S5 = 5a1 + 10d = 1/5 (1)
S10 = 10a1 + 45d = 1/10 (2)
Multiply (1) through by -2 and add it to (2)
25d = -3 /10
d = -3 / 250
5a1 + 10 (-3/250) = 1/5
5a1 + -30/250 = 1/5
5a1 = 1/5 + 3/25
5a1 = 8/25
a1 = 8/125
S15 = 15 ( a1) + 105 d = 15(8/125) + 105 ( -3/250) = -3/10
