Question

If the expansion in powers of x of the function `1/((1 - ax)(1 - bx))` is a₀ + a₁x + a₂x² + a₃x³ ....... then a_n is ______.

Options

• `(b^n - a^n)/(b - a)`

• `(a^n - b^n)/(b - a)`

• `(a^(n + 1) - b^(n + 1))/(b - a)`

• `(b^(n + 1) - a^(n + 1))/(b - a)`

Answer 1

If the expansion in powers of x of the function `1/((1 - ax)(1 - bx))` is a₀ + a₁x + a₂x² + a₃x³ ....... then a_n is `underlinebb((b^(n + 1) - a^(n + 1))/(b - a))`.

Explanation:

(1 – ax)^–1(1 – bx)^–1

= (1 + ax + a²x² + ...)(1 + bx + b²x² + ...)

∴ Coefficient of xⁿ

= xⁿ = bⁿ + ab^n–1 + a²b^n–2 + ....... + a^n–1b + aⁿ

which is a G.P. with r = `a/b`

 ∴ Its sum is = `(b^n[1 - (a/b)^(n + 1)])/(1 - a/b)`

= `(b^(n + 1) - a^(n + 1))/(b - a)`

∴ a_n = `(b^(n + 1) - a^(n + 1))/(b - a)`