Show that the middle term in the expansion of 1 + x 2 n is 1 · 3 · 5 … … … ( 2 n - 1 ) n ! 2 n x n .

In the expansion of 1 + x 2 n , the middle term will be 2 n + 2 2 = n + 1 th term ∴ T n + 1 = C n 2 n · x n = ( 2 n ) ! n ! × n ! · x n = 1 . 2 · 3 · 4 · 5 · 6 · … · ( 2 n - 2 ) ( 2 n - 1 ) · 2 n n ! × n ! · x n . = 1 · 3 · 5 · 7 · 9 · … · ( 2 n - 1 ) n ! × 2 · 4 · 6 · ⋯ · · 2 n n ! × x n = 1 · 3 · 5 … … ( 2 n - 1 ) n ! · 2 n x n

If C 0 , C 1 , C 2 , … … . , C n denote the coefficient in the expansion of ( 1 + x ) n , prove that, C 1 C 0 + 2 C 2 C 1 + 3 C 3 C 2 + 4 C 4 C 3 + … … … + n C n C n - 1 = n ( n + 1 ) 2 .

L. H. S. = C 1 C 0 + 2 C 2 C 1 + 3 C 3 C 2 + … … . . n C n C n - 1 We have, C 1 C 0 = n 1 = n 2 C 2 C 1 = 2 n ( n - 1 ) n · 2 ! = n - 1 3 C 3 C 2 = 3 n ( n - 1 ) ( n - 2 ) 3 ! × 2 ! n ( n - 1 ) = n - 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . and n C n C n - 1 = n n = 1 . Adding these all, we get C 1 C 0 + 2 C 2 C 1 + 3 C 3 C 2 + … . + n C n C n - 1 = n + ( n - 1 ) + ( n - 2 ) + … + 3 + 2 + 1 = 1 + 2 + 3 + 4 + … . + ( n - 2 ) + ( n - 1 ) + ( n ) = ∑ r = 1 n r = n 2 ( n + 1 ) .

If C 0 , C 1 , C 2 , … … … . , C n denote the coefficients in the expansion of 1 + x n , prove that 2 C 0 + 2 2 C 1 2 + 2 3 C 2 3 + 2 4 C 3 4 + … + 2 n + 1 C n n + 1 = 3 n + 1 - 1 n + 1 .

C 0 , C 1 , C 2 denote the coefficients in the expansion of 1 + x n . The expansion of 1 + x n = 1 + C 1 n x + C 2 n x 2 + … + C n n x n = C 0 + C 1 x + C 2 x 2 + … + C n x n From the property, we can write C r n = C n - r n So, C 0 = C n = 1 , C 1 = C n - 1 = n , C 2 = C n - 2 = n ! 2 ! × n - 2 ! = n × n - 1 2 ! . Now, we have 2 C 0 + 2 2 C 1 2 + 2 3 C 2 3 + 2 4 C 3 4 + … … + 2 n + 1 C n n + 1 = 2 · 1 + 2 2 · n 2 + 2 3 3 · n ( n - 1 ) 2 ! + … … … + 2 n + 1 × 1 n + 1 = 1 n + 1 × 2 × n + 1 + n ( n + 1 ) 2 ! · 2 2 + … + 2 n + 1 = 1 n + 1 × 1 + n + 1 · 2 + n ( n + 1 ) 2 ! · 2 2 + … + 2 n + 1 - 1 = 1 n + 1 × 1 + 2 n + 1 - 1 = 1 n + 1 × 3 n + 1 - 1 = 3 n + 1 - 1 n + 1

E M B I B E

Higher Algebra>Chapter 13 - Binomial Theorem: Positive Integral Index>EXAMPLES. XIII.b.

H S Hall and S R Knight Solutions for Chapter: Binomial Theorem: Positive Integral Index, Exercise 2: EXAMPLES. XIII.b.

Author:H S Hall & S R Knight

H S Hall Mathematics Solutions for Exercise - H S Hall and S R Knight Solutions for Chapter: Binomial Theorem: Positive Integral Index, Exercise 2: EXAMPLES. XIII.b.

Attempt the practice questions on Chapter 13: Binomial Theorem: Positive Integral Index, Exercise 2: EXAMPLES. XIII.b. with hints and solutions to strengthen your understanding. Higher Algebra solutions are prepared by Experienced Embibe Experts.

Questions from H S Hall and S R Knight Solutions for Chapter: Binomial Theorem: Positive Integral Index, Exercise 2: EXAMPLES. XIII.b. with Hints & Solutions

EASY

JEE Main

IMPORTANT

Higher Algebra>Chapter 13 - Binomial Theorem: Positive Integral Index>EXAMPLES. XIII.b.>Q 16

Show that the middle term in the expansion of $(1 + x)^{2 n}$ is $\frac{1 \cdot 3 \cdot 5 \dots \dots \dots (2 n - 1)}{n!} 2^{n} x^{n}$ .

EASY

JEE Main

IMPORTANT

Higher Algebra>Chapter 13 - Binomial Theorem: Positive Integral Index>EXAMPLES. XIII.b.>Q 17

If $C_{0}, C_{1}, C_{2}, \dots \dots \dots C_{n}$ denote the coefficients in the expansion of $(1 + x)^{n} .$ Prove that $C_{1} + 2 C_{2} + 3 C_{3} + \dots \dots + n C_{n} = n 2^{n - 1}$ .

EASY

JEE Main

IMPORTANT

Higher Algebra>Chapter 13 - Binomial Theorem: Positive Integral Index>EXAMPLES. XIII.b.>Q 18

If $C_{0}, C_{1}, C_{2}, C_{3}, \dots \dots . C_{n}$ denote the coefficients in the expansion of $(1 + x)^{n},$ prove that, $C_{0} + \frac{C_{1}}{2} + \frac{C_{2}}{3} + \frac{C_{3}}{4} + \dots \dots + \frac{C_{n}}{n + 1} = \frac{2^{n + 1} - 1}{n + 1}$ .

EASY

JEE Main

IMPORTANT

Higher Algebra>Chapter 13 - Binomial Theorem: Positive Integral Index>EXAMPLES. XIII.b.>Q 19

If $C_{0}, C_{1}, C_{2}, \dots \dots ., C_{n}$ denote the coefficient in the expansion of $(1 + x)^{n},$ prove that, $\frac{C_{1}}{C_{0}} + \frac{2 C_{2}}{C_{1}} + \frac{3 C_{3}}{C_{2}} + \frac{4 C_{4}}{C_{3}} + \dots \dots \dots + \frac{n C_{n}}{C_{n - 1}} = \frac{n (n + 1)}{2}$ .

EASY

JEE Main

IMPORTANT

Higher Algebra>Chapter 13 - Binomial Theorem: Positive Integral Index>EXAMPLES. XIII.b.>Q 20

If $C_{0}, C_{1}, C_{2}, C_{3}, \dots \dots . C_{n}$ denote the coefficients in the expansion of $(1 + x)^{n},$ prove that $(C_{0} + C_{1}) (C_{1} + C_{2}) (C_{2} + C_{3}) \dots (C_{n - 1} + C_{n}) = \frac{C_{1} C_{2} C_{3} \dots C_{n} (n + 1)^{n}}{n!}$ .

HARD

JEE Main

IMPORTANT

Higher Algebra>Chapter 13 - Binomial Theorem: Positive Integral Index>EXAMPLES. XIII.b.>Q 21

If $C_{0}, C_{1}, C_{2}, \dots \dots \dots ., C_{n}$ denote the coefficients in the expansion of ${(1 + x)}^{n},$ prove that $2 C_{0} + \frac{2^{2} C_{1}}{2} + \frac{2^{3} C_{2}}{3} + \frac{2^{4} C_{3}}{4} + \dots + \frac{2^{n + 1} C_{n}}{n + 1} = \frac{3^{n + 1} - 1}{n + 1}$ .

EASY

JEE Main

IMPORTANT

Higher Algebra>Chapter 13 - Binomial Theorem: Positive Integral Index>EXAMPLES. XIII.b.>Q 22

If $C_{0}, C_{1}, C_{2}, \dots, C_{n}$ denotes the coefficients in the expansion of $(1 + x)^{n},$ prove that $C_{0}^{2} + C_{1}^{2} + C_{2}^{2} + \dots . + C_{n}^{2} = \frac{2 n!}{n! n!}$ .

EASY

JEE Main

IMPORTANT

Higher Algebra>Chapter 13 - Binomial Theorem: Positive Integral Index>EXAMPLES. XIII.b.>Q 23

If $C_{0}, C_{1}, C_{2}, \dots . ., C_{n}$ denote the coefficients in the expansion of $(1 + x)^{n},$ prove that, $C_{0} C_{r} + C_{1} C_{r - 1} + C_{2} C_{r - 2} + \dots + C_{n - r} C_{n} = \frac{2 n!}{(n - r)! (n + r)!}$

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