Osman A. answered • 09/21/21

Professor of Business Mathematics – Business Math/Finite Math

The sum of the first five terms in an arithmetic sequence is −40. If 𝑎9=16, find the common difference.

S_{5} = -40 a_{9} = 16

a_{n} = a_{1} + (n - 1)d

a_{9} = a_{1} + (9 - 1)d ==> 16 = a_{1} + 8d ==> a_{1} = 16 - 8d

S_{n} = (n/2)(2a_{1} + (n - 1)d)

S_{5} = (5/2)(2a_{1} + (5 - 1)d) ==> -40 = (5/2)(2a_{1} + 4d) ==> (-40)(2/5) = 2a_{1} + 4d ==> -16 = 2a_{1} + 4d

-16 = 2a_{1} + 4d ==> -16 = 2(16 - 8d) + 4d ==> -16 = 32 - 16d + 4d ==> -16 = 32 - 12d ==> 12d = 32 + 16

12d = 48 ==> d = 48/12 = 4

d = 4 (Final Answer)