Question: find n with first term -3, sum is 31.5 and common difference is 2.5 .
A1 = -3, Sn = 31.5, d = 2.5
In any arithmetic sequence, the sum of the 1st n terms is given by
Sn = n(2*A1 + (n-1)d)/2
Using the information given and the formula for the sum of 1st n terms,
31.5 = n(2*(-3) + (n-1)*2.5)/2
63 = n(-6 + 2.5(n-1))
126 = n(-12 + 5n - 5)
126 = -17n + 5n^2
5n^2 - 17n - 126 = 0
(5n + 18)(n - 7) = 0
n = -18/5, n = 7 (but n must be a positive integer)
n = 7
Therefore, Answer: n = 7