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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.

 Jun 27, 2024
 #1
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We can set variables up to solve this problem. Let's set \(d = a_2 -a_1\)

 

From the problem, we can write the equations

\(S_5 = 5a_1 + (d + 2d + 3d + 4d) = 5a_1 + 10d\\S_{10} = 10a_1 + (d + 2d + \cdots + 9d) = 10a_1 + 45d\\\\\S_{15} = 15a_1 + (d + 2d + \cdots + 14d) = 15a_1 + 105d\)

 

Simplifying the first 2 equations, we get the conditions

\(\begin{cases} 5a_1 + 10d = \dfrac15\\ 10a_1 + 45d = \dfrac1{10} \end{cases}\)

 

Solving this gives that d is -3/250 and a_1 is 8/125. 

 

Thus, we have

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

 

So -3/10 is our answer. 

 

Thanks! :)

 Jun 27, 2024

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