The 5th and 10th terms of an arithmetic progression are 18 and -2 respectively.

Question

(a) Find the common difference and the first term.
(b) Determine the least number of terms which must
be added together so that the sum of the progression is negative .Hence find the sum.

Answer 1

Using our definitions:
a + 4d = 18
a + 9d = -2
subtract:
5d = -20
d = -4
mentally in a + 4d = 18, a = 34 <----- first term

so we want S(n) < 0, let's see when it is zero
(n/2)(2a + (n-1)d) = 0
n/2)(68 - 4(n-1)) = 0
n/2(68 - 4n + 4) = 0
n(36 - 2n) = 0
-2n^2 + 36n = 0
n^2 - 18n = 0
n = 0 or n = 18

So when we add the first 18 terms we get zero, the terms are getting smaller
so when we add 19 terms we get a negative sum