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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 14, 2024

Best Answer 

 #1
avatar+1280 
+1

Let's set some variables and make some observations. 

Let's let \(d = a_2 -a_1\)

 

Now, let's note something important. Note that

\(S_5 = 5a_1 + (d + 2d + 3d + 4d) = 5a_1 + 10d\)

\(S_{10} = 10a_1 + (d + 2d + \cdots + 9d) = 10a_1 + 45d\)

\(S_{15} = 15a_1 + (d + 2d + \cdots + 14d) = 15a_1 + 105d\)

 

These observations are really important for us to solve the problem. 

Let's take two of these cases. We can write a system of equations, and we get

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

 

Now, we solve for a1 and d. 

Solving this system of equations, we get

\(d = -\dfrac3{250}\\ a_1 = \dfrac8{125}\)

 

Plugging this into S15, we get

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

 

So -3/10 is our final answer. 

 

Thanks! :)

 Jun 14, 2024
 #1
avatar+1280 
+1
Best Answer

Let's set some variables and make some observations. 

Let's let \(d = a_2 -a_1\)

 

Now, let's note something important. Note that

\(S_5 = 5a_1 + (d + 2d + 3d + 4d) = 5a_1 + 10d\)

\(S_{10} = 10a_1 + (d + 2d + \cdots + 9d) = 10a_1 + 45d\)

\(S_{15} = 15a_1 + (d + 2d + \cdots + 14d) = 15a_1 + 105d\)

 

These observations are really important for us to solve the problem. 

Let's take two of these cases. We can write a system of equations, and we get

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

 

Now, we solve for a1 and d. 

Solving this system of equations, we get

\(d = -\dfrac3{250}\\ a_1 = \dfrac8{125}\)

 

Plugging this into S15, we get

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

 

So -3/10 is our final answer. 

 

Thanks! :)

NotThatSmart Jun 14, 2024

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