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Find the 31st term of an arithmetic sequence where the eighth term is 45 and the fourth term is 65.

User Lest
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Let a(n) denote the nth term in the sequence. Then

a(5) = a(4) + d

a(6) = a(5) + d = a(4) + 2d

a(7) = a(6) + d = a(4) + 3d

a(8) = a(7) + d = a(4) + 4d

where d is the fixed difference between consecutive terms in the sequence.

We have a(4) = 65 and a(8) = 45, so we solve for d:

45 = 65 + 4d ==> d = -5

so, given some term in the sequence a(n), the next term is a(n + 1) = a(n) - 5.

Continuing the pattern above, we would wind up getting

a(31) = a(4) + 27d

(In case the pattern isn't clear, watch the coefficient of d; then 1 + 4 = 5, 2 + 4 = 6, 3 + 4 = 7, and so on.)

Then the 31st term in the sequence must be

a(31) = 65 + 31(-5) = -90

User EthanK
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