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 $x^{3}$ .........

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

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

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

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

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Solution

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

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

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Match the following (|x| < 1) (1) (1+x)−1 (P) 1 + 2x + 3 x2 + 4 x3......... (2) (1−x)−1 (Q) 1 - x + x2 - x3......... (3) (1+x)−2 (R) 1 + x + x2 + x3......... (4) (1−x)−2 (S)1 - 2x + 3 x2 - 4 x3.........

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 $x^{3}$ .........