Question

If the sum of first $9$ terms of an A.P. is $72$ and the common difference is $5$, find the first term and the $10th$ term of the A.P.

Answer 1

The first number is x, the second is x+5, the 3rd is x+10, etc. So the sum of the first 9 terms is 9x + (5 + 10 + 15 + etc.) = 9x + 5(1 + 2 + 3 + 4 + 5 + 6 + 7 + 8). Since the sum of the first n positive integers is n(n+1)/2, The sum of the first 9 terms is

9x + 5(8*9/2) = 9x + 180.

So, 9x + 180 = 72. Therefore, x = (72–180)/9 = -12. The 10th term is -12 + 5*9 = 33.