Use the formula for the sum of the first n integers and/or the formula for the sum of a geometric sequence to evaluate summations and write them in a closed form.

Question

Use the formula for the sum of the first n integers and the formula

for the sum of a geometric sequence to evaluate

(7^2 − 5) + (7^3 − 10) + (7^4 − 15) + · · · + (7^2023 − 10110).

Do not compute powers of 7 for this problem. Instead, write your answer as

simply as possible without computing powers of 7. Your answer may involve 7x

for some value(s) of x.

Hint: You can rewrite this summation as a combination of two summations.

You must use the fact that for every integer n ≥ 1,

1+2+ ... n = n(n+1)/2,

and the fact that for every real number r except 1, and any integer n ≥ 0,\

n

∑ r^j = r^n+1 - 1 / r-1

.i=1

Answer 1

(7^2 − 5) + (7^3 − 10) + (7^4 − 15) + · · · + (7^2023 − 10110)

5+10 + 15 + ... +10110 = 5(1+ 2 + ... +2022) = ...

7^2 +7^3 + ...+7^2023 = 7^2(7^0 + 7^1 + ... + 7^2021) = ...