If C 0 , C 1 , C 2 , … C n are the binomial coefficients in the expansion of 1 + x n , then prove that: 3 · 2 - 1 2 C 1 + 3 2 · 2 2 - 1 2 2 C 2 + 3 3 · 2 3 - 1 2 3 C 3 + … … . + 3 n · 2 n - 1 2 n C n = 2 3 n - 3 n 2 n

We know, 1 + x n = ∑ r = 0 n x r C r Given, 3 · 2 - 1 2 C 1 + 3 2 · 2 2 - 1 2 2 C 2 + 3 3 · 2 3 - 1 2 3 C 3 + … … . + 3 n · 2 n - 1 2 n C n = ∑ r = 0 n 3 r C r - ∑ r = 0 n 1 2 r C r = 1 + 3 n - 1 + 1 2 n = 2 3 n - 3 n 2 n

E M B I B E

Alpha Question Bank for Engineering: Mathematics>Chapter 7 - Binomial Theorem>Exercise-1>Q 15

EASY

JEE Main/Advance

IMPORTANT

Earn 100

Using binomial theorem prove that $6^{n} - 5 n$ always leaves the remainder $1$ when divided by $25$ .

Important Questions on Binomial Theorem

EASY

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 7 - Binomial Theorem>Exercise-1>Q 16

Find the last digit, last two digits and last three digits of the number

{(27)}^{27}

.

MEDIUM

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 7 - Binomial Theorem>Exercise-1>Q 18

Find numerically greatest term(s) in the expansion of

{(3 - 5 x)}^{15}

when

x = \frac{1}{5}

.

MEDIUM

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 7 - Binomial Theorem>Exercise-1>Q 19

Which term is numerically the greatest term in the expansion of

{(2 x + 5 y)}^{34},

when

x = 3

and

y = 2 ?

EASY

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 7 - Binomial Theorem>Exercise-1>Q 20

Find the term in the expansion of

{(2 x - 5)}^{6}

which have
(i) Greatest binomial coefficient
(ii) Greatest numerical coefficient
(iii) Algebraically greatest coefficient
(iv) Algebraically least coefficient

EASY

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 7 - Binomial Theorem>Exercise-1>Q 21

If $C_{0}, C_{1}, C_{2}, \dots C_{n}$ are the binomial coefficients in the expansion of ${(1 + x)}^{n}$ , then prove that:

$\frac{(3 \cdot 2 - 1)}{2} C_{1} + \frac{3^{2} \cdot 2^{2} - 1}{2^{2}} C_{2} + \frac{3^{3} \cdot 2^{3} - 1}{2^{3}} C_{3} + \dots \dots . + \frac{3^{n} \cdot 2^{n} - 1}{2^{n}} C_{n} = \frac{2^{3 n} - 3^{n}}{2^{n}}$

EASY

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 7 - Binomial Theorem>Exercise-1>Q 22

If $C_{0}, C_{1}, C_{2}, \dots C_{n}$ are the binomial coefficients in the expansion of ${(1 + x)}^{n}$ then prove that

$\frac{C_{1}}{C_{0}} + 2 \frac{C_{2}}{C_{1}} + 3 \frac{C_{3}}{C_{2}} + \dots \dots . . + n \frac{C_{n}}{C_{n - 1}} = \frac{n (n + 1)}{2}$

EASY

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 7 - Binomial Theorem>Exercise-1>Q 23

If $C_{0}, C_{1}, C_{2}, \dots C_{n}$ are the binomial coefficients in the expansion of ${(1 + x)}^{n}$ then prove that:

$(C_{0} + C_{1}) (C_{1} + C_{2}) (C_{2} + C_{3}) (C_{3} + C_{4}) \dots \dots . (C_{n - 1} + C_{n}) = \frac{C_{0} C_{1} C_{2} \dots \dots \dots C_{n - 1} {(n + 1)}^{n}}{n!}$

EASY

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 7 - Binomial Theorem>Exercise-1>Q 24

If $C_{0}, C_{1}, C_{2}, \dots C_{n}$ are the binomial coefficients in the expansion of ${(1 + x)}^{n}$ then prove that

$C_{0} - 2 C_{1} + 3 C_{2} - 4 C_{3} + \dots . + {(- 1)}^{n} (n + 1) C_{n} = 0$