CAT 2002 QA Question 22 · Algebra - Progressions

S = 2x + 5x² + 9x³ + 14x⁴ + 20x⁵ + ...                              …(i)

Multiplying both sides by x, we get

xS = 2x² + 5x³ + 9x⁴ + 14x⁵ + 20x⁶ + ...                             ...(ii)

Subtracting (ii) from (i),

(1 – x)S = 2x + 3x² + 4x³ + 5x⁴ + 6x⁵ + ...                         ...(iii)

Again multiplying both the sides by x, we get

x(1 – x)S = 2x² + 3x³ + 4x⁴ + 5x⁵ + 6x⁶ + ...                      ...(iv)

Subtracting (iv) from (iii),

(1 – x)S – x(1 – x)S = 2x + x² + x³ + x⁴ + x⁵ + ...

(1 – x)²S = 2x + x² + x³ + x⁴ + x⁵ +...

(1 – x)²S = x + (x + x² + x³ + ... + ∞)

(1 - x)²S = x + $\frac{x}{1-x}$ ∵ S_∞ = $\frac{a}{1-r}$ for r < 1

∴ (1 - x)²S = $\frac{x-x^2+x}{1-x}$

∴ S = $\frac{x(2-x)}{{(1-x)}^3}$

Hence, option (a).