math precalculus 12

Question

Given the Arithmetic series with S₆ = -15, S₇ = -28, and d = -3 find S₄.

Answer 1

S₇ - S₆ = the 7th term. (since the only thing different about the two sums is adding in the 7th term for S₇)

So t₇ = - 13 and since we also know d = -3 we can work backwards to get t₁ , then S₄:

t₇ = t₁ -3(7-1) = - 13 and t₁ = 5. (Easy to verify: 5 , 2 , -1, -4, -7, -10 , -13) (The sums are correct too.)

By inspection, S₄ = 2. Or, using the formula, S₄ = (4/2)(5 + -4) = 2

If you prefer, you can skip finding t₁ once you know t₇ = -13: t₆ = -10 and t₅ = -7 . Then S₄ = S₆ - t₆ - t₅