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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 \(S_{15}.\)

 Apr 9, 2024
 #1
avatar+128794 
+1

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

 

cool cool cool

 Apr 9, 2024

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