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The sum of the first three terms of an arithmetic sequence is 39, and the sum of the last four terms is 178.  If the first term is 10, then what is the average of all the terms?

 Feb 19, 2021
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Call the first  three terms   10  , 10 + d , 10 +  2d

 

So.....

 

10   +  (10  + d)  +  (10 + 2d)   = 39     simplify

 

30  +  3d   =  39

 

3d  =  39 -  30

 

3d    = 9

 

d   = 3

 

Call   the  last  4  terms      n , n + 3, n + 6 , n + 9

 

So

 

n  +  (n + 3) + ( n + 6)  + ( n + 9)  =  178

 

4n +  18    =  178

 

4n   =  178  -  18

 

4n   = 160

 

n =  40

 

The first term is  10     and  the last  term  =   40  + 9 =  49 =    10  + 3 (14  - 1)

 

There  are   14  terms

 

Their sum    =  [  first term  +  last term ]  *  number  of terms    /  2  =    [ 10  + 49] * 14/2   =  413

 

The  average  =     413 /  14    =    29.5

 

cool cool cool

 Feb 19, 2021
edited by CPhill  Feb 19, 2021

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