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?
+128474 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
