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Let a1, a2, a3, ... be an arithmetic sequence.  Let Sn denote the sum of the first n terms.  If S_5 = 1/5 and S_10 = 1/10, then find S_15.

 May 13, 2024
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Let \(d = a_2 -a_1\). Note that \(S_5 = 5a_1 + (d + 2d + 3d + 4d) = 5a_1 + 10d\), and \(S_{10} = 10a_1 + (d + 2d + \cdots + 9d) = 10a_1 + 45d\), and \(S_{15} = 15a_1 + (d + 2d + \cdots + 14d) = 15a_1 + 105d\).

 

So, the given conditions are \(\begin{cases} 5a_1 + 10d = \dfrac15\\ 10a_1 + 45d = \dfrac1{10} \end{cases}\). Solving gives \(d = -\dfrac3{250}\) and \(a_1 = \dfrac8{125}\).

 

So \(S_{15} = 15a_1 + 105d = -\dfrac3{10}\).

 May 13, 2024

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