Question

Obtain the binomial expansion of (2-x) (1+1/2x)^8 in ascending powers of x as far as the term in x^3. Use your result to estimate the value of 1.9 times (1.05)^8.

Answer 1

(2-x) (1+1/2 x)^8
= (2-x)[1 + 8(1/2)x)) + 28(1/4)x^2 + 56(1/8)x^3 + ....]
= (2-x)(1 + 4x + 7x^2 + 7x^3 + )
= 2 + 8x + 14x^2 + 14x^3 + ... - x - 4x^2 - 7x^3
= appr 2 + 7x + 10x^2 + 7x^3

so comparing
1.9(1.05)^8 with (2-x)(1 + 1/2 x)^8
x = .1

I get :
2 + .7 + 10(.01) + 7(.001)
= 2 + .7 + .1 + .007
= 2.807

actual answer by calculator is 2.807165343

NOTE:
A rather purely academic exercise.
"Today" we just use a calculator.