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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 $S_{20} = \frac{1}{5}$ and $S_{10} = 0,$ then find $S_{70}.$

 Jan 11, 2024
 #1
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Let the first term   =  S1    and the  common  difference = d

Sum of first n terms =  n *S1  + [ (n -1) (n) / 2  ] d

S20  =   20S1  + 190 d =  1/5   (1)

S10  =   10S1  +   45d =   0     (2)

 

Multiply  (1)  by -2

 

-20S1 - 90d  =  0         add this to  (1)

 

100d  =  1/5

 

d =  1/500

 

10S1  + (45)(1/500)   = 0

 

10S1  =  -45 / 500   =  -9/100

 

10S1  = -9/100

 

S1  =  -9 / 1000

 

 

S70  =  70S1  + 2415d  =    70 (-9/1000)  + 2415 (1/500)  =   21 / 5  

 

cool cool cool

 Jan 11, 2024

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