C 0 - 3 2 - C 1 - 9 3 - C 2 - 27 4 - C 3 - 3 n n-1 - C n - 4 n-1 -1 3 n-1 -Is it true -If true enter 1 else 0-

Solve: we have Given. c_0+3/2· c_1+9/3 c_2+27/4 c_3+⋯+3^n/n+1 c_n=4^n+1 1/3(n+1) By binomial expansion (1+x)^n=^nc_0+^nc_1 x+^n c_2 x^2+… . .^n c_n x^n ...

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Byju's Answer

Standard XII

Mathematics

General Term of Binomial Expansion

C 0 + 3 2 . C...

Question

$C_{0} + \frac{3}{2} . C_{1} + \frac{9}{3} . C_{2} + \frac{27}{4} . C_{3} + . . . . . . . . . . . + \frac{3^{n}}{n + 1} . C_{n} = \frac{4^{n + 1} - 1}{3 (n + 1)}$ .Is it true ?If true enter 1 else 0.

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Solution

Solve: we have Given.

$c_{0} + \frac{3}{2} \cdot c_{1} + \frac{9}{3} c_{2} + \frac{27}{4} c_{3} + \dots + \frac{3^{n}}{n + 1} c_{n} = \frac{4^{n + 1} - 1}{3 (n + 1)}$

By binomial expansion

$(1 + x)^{n} =^{n} c_{0} +^{n} c_{1} x +^{n} c_{2} x^{2} + \dots . .^{n} c_{n} x^{n}$

on integrating both sides with respect to $x$ we get

$\int_{0}^{x} (1 + x)^{n} d x = \int (^{n} c_{0} +^{n} c_{1} x +^{n} c_{2} x^{2} + \dots^{n} c_{n} x^{n}) d x$

$\Rightarrow \frac{(1 + x)^{n + 1} - 1}{n + 1} =^{n} c_{0} x +^{n} c_{1} \frac{x^{2}}{2} +^{n} c_{2} \frac{x^{3}}{3} + \dots^{n} c_{n} \frac{x^{n + 1}}{n + 1}$

on putting $x = 3$ we get

$\Rightarrow \frac{4^{n + 1} - 1}{n + 1} = 3^{n} c_{0} + \frac{9}{2}^{n} c_{1} + \frac{27}{3}^{n} c_{2} + \dots . . \frac{3^{n + 1}}{n + 1}^{n} c_{n}$

$\Rightarrow \frac{4^{n + 1} - 1}{n + 1} = 3 [^{n} c_{0} + \frac{3}{2}^{n} c_{1} + \frac{9}{3}^{n} c_{2} + \dots \dots \frac{3^{n + 1}}{n + 1}^{n} c_{n}]$

$\Rightarrow^{n} c_{0} + \frac{3}{2}^{n} c_{1} + \frac{9}{3}^{n} c_{2} + \dots \frac{3^{n}}{n + 1} = \frac{4^{n + 1} - 1}{3 (n + 1)}$

Hence It is correct
so, correct answer is 1 .

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Similar questions

C_{0} + \frac{3}{2} . C_{1} + C_{2} + \frac{27}{4} . C_{3} + + \frac{3^{n}}{n + 1} . C_{n} = \frac{4^{n + 1} + 1}{3 (n + 1)}

Q. Prove that

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

Q. The value of

2 (^{n} C_{0}) + \frac{3}{2} (^{n} C_{1}) + \frac{4}{3} (^{n} C_{2}) + \frac{5}{4} (^{n} C_{3}) . . .

is:

\frac{C_{1}}{C_{0}} + 2. \frac{C_{2}}{C_{1}} + 3. \frac{C_{3}}{C_{2}} + . . . . . + n . \frac{C_{n}}{C_{n - 1}}

Q. The sum of the series

2 C_{0} + \frac{C_{1}}{2} \cdot 2^{2} + \frac{C_{2}}{3} \cdot 2^{3} + \frac{C_{3}}{4} \cdot 2^{4} + \dots + \frac{C_{n}}{n + 1} \cdot 2^{n + 1}

is equal to , where

(C_{r} =^{n} C_{r})

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C0+32.C1+93.C2+274.C3+...........+3nn+1.Cn=4n+1−13(n+1).Is it true ?If true enter 1 else 0.

$C_{0} + \frac{3}{2} . C_{1} + \frac{9}{3} . C_{2} + \frac{27}{4} . C_{3} + . . . . . . . . . . . + \frac{3^{n}}{n + 1} . C_{n} = \frac{4^{n + 1} - 1}{3 (n + 1)}$ .Is it true ?If true enter 1 else 0.