What is the result of the sum (5-2a+3b)+(5-2a+3b)^2+...+(5-2a+3b)^500 if 2a-3b=6 ?

The value of 2a - 3b = 6

We
expression we have to find the sum of is (5- 2a + 3b)+(5- 2a + 3b)^2+...+(5 - 2a +
3b)^500

(5- 2a + 3b)+(5- 2a + 3b)^2+...+(5 - 2a +
3b)^500

=> (5- (2a - 3b))+(5- (2a - 3b))^2+...+(5 -
(2a - 3b))^500

=> (5- 6)+(5- 6)^2+...+(5 -
6)^500

=> -1 + -1^2 + -1^3 +...+
-1^500

=> -1 + 1 - 1 + 1 - 1 +... +
1

The exponents of -1 are from 1 to 500. From 1 to 500 we
an equal number of even and odd
numbers.

Therefore the sum of the terms is
equal to 0. We get (5- (2a - 3b))+(5- (2a - 3b))^2+...+(5 - (2a - 3b))^500 =
0