Question

Match the following (|x| < 1)

(1) ${(1+x)}^{-1}$ (P) 1 + 2x + 3 $x^2$ + 4 $x^3$.........

(2) ${(1-x)}^{-1}$ (Q) 1 - x + $x^2$ - $x^3$.........

(3) ${(1+x)}^{-2}$ (R) 1 + x + $x^2$ + $x^3$.........

(4) ${(1-x)}^{-2}$ (S)1 - 2x + 3 $x^2$ - 4 $x^3$.........

• A. 1-R, 2-Q, 3-S, 4-P
• B. 1-Q, 2-R, 3-P, 4-S
• C. 1-R, 2-S, 3-Q, 4-P
• D. 1 - Q, 2 - R, 3 - S, 4 - P

Answer 1

The correct option is D

1 - Q, 2 - R, 3 - S, 4 - P

If n is a rational number and |x| < 1,

${(1+x)}^n$ = 1 + nx + $\frac{n(n-1)}{2!}$ $x^2$ + $\frac{n(n-1)(n-2)}{3!}$ $x^3$........

(1) n = -1

$(1+x{)}^{-1}=1-x+\frac{(-1)(-2)x^2}{2!}+\frac{(-1)(-2)(-3)}{3!}{x}^3...$

= 1 - x + $x^2$ - $x^3$.........

(2) ${(1-x)}^{-1}$ = 1 + (-1)(-x) + $\frac{(-1)(-2)}{2!}$ × ${(-x)}^2$ + $\frac{(-1)(-2)(-3)}{3!}$ × ${(-x)}^3$.........

= 1 + x + $x^2$ + $x^3$......

(3) ${(1+x)}^{-2}$ = 1 + (-2)x + $\frac{(-2)(-3)}{2!}$ × $x^2$ + $\frac{(-2)(-3)(-4)}{3!}$ × $x^3$.........

= 1 - 2x + 3$x^2$ - 4 $x^3$......

(4) $(1-x{)}^{-2}=1+(-2\left)\right(-x)+\frac{(-2)(-3)(-x{)}^2}{2!}+\frac{(-2)(-3)(-4)}{3!}\times (-x{)}^3....$

= 1 + 2x + 3$x^2$ + 4 $x^3$......