Question

What is the fourth term of the expansion of the binomial (2x+5)^5

Answer 1

For a binomial expansion with exponent 5, the general equation is as follows
(x + y)⁵ = x⁵ + 5x⁴y + 10x³y² + 10x²y³ + 5xy⁴ + y⁵

when expanded from left to right the powers of x decrease from 5 to 0 and powers of y increase from 0 to 5

substitute x =2x and y = 5

(2x + 5)⁵ = (2x)⁵ + 5(2x)⁴5 + 10(2x)³5² + 10(2x)²5³ + 5(2x)5⁴ + 5⁵

              = 32x⁵ + 400x⁴ + 2000x³ + 5000x² + 6250x + 3125
we use the pascals triangle to find the coefficient

we can use the binomial theorem to find the 4th term as well
exponent - 5
term - 4
5!/ (5-4)!4! * (2x)⁴5⁵⁻⁴
=120/24 *16x⁴*5
=400x⁴