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

Answer 1

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

S₅ = -40 a₉ = 16

a_n = a₁ + (n - 1)d

a₉ = a₁ + (9 - 1)d ==> 16 = a₁ + 8d ==> a₁ = 16 - 8d

S_n = (n/2)(2a₁ + (n - 1)d)

S₅ = (5/2)(2a₁ + (5 - 1)d) ==> -40 = (5/2)(2a₁ + 4d) ==> (-40)(2/5) = 2a₁ + 4d ==> -16 = 2a₁ + 4d

-16 = 2a₁ + 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)

Answer 2

The sum of n consecutive items in a sequence is the middle number times the number of terms (the other terms balance out to the middle).

Thus, the middle term of the first five terms is a3, and 5a3 = -40.

a3 = -8

a9 = 16

d = (16 - -8)/(9 - 3) = 24/6 = -4