1. Determine the sum of the first 53 terms of the following series: 179+173+167+... 2. Determine the sum of the first 19 terms of the following series: 6−12+24−48+...

(1) This series consists of terms of an arithmetic sequence:

179 - 173 = 6

173 - 167 = 6

and so on, so that the n-th term in the series is (for n ≥ 1)

a(n) = 179 - 6 (n - 1) = 185 - 6n

Then the sum of the first 53 terms is

(2) This series has terms from a geometric sequence:

-12 / 6 = -2

24/(-12) = -2

-48/24 = -2

and so on. The n-th term is (again, for n ≥ 1)

a(n) = 6 (-2)ⁿ⁻¹

and the sum of the first 19 terms is

Multiply both sides by -2 :

Subtracting this from the first sum gives

and solving for the sum, you get

answer:

sometimes

*i think