JAMIA 2022 — Mathematics PYQ

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Question

JAMIA 2022 — Mathematics PYQ

JAMIA | Mathematics | 2022

In the expansion of (1+x)⁵⁰ the sum of coefficients of odd powers of x is

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Explanation

1. General Expansion

The binomial expansion of $(1+x)^n$ is:

$(1+x)^n = C_0 + C_1x + C_2x^2 + C_3x^3 + \dots + C_nx^n$

2. Finding the Sum of All Coefficients

Put $x = 1$ :

$(1+1)^{50} = C_0 + C_1 + C_2 + C_3 + \dots + C_{50}$

$2^{50} = C_0 + C_1 + C_2 + C_3 + \dots + C_{50}$ --- (Equation 1)

3. Finding the Alternating Sum

Put $x = -1$ :

$(1-1)^{50} = C_0 - C_1 + C_2 - C_3 + \dots + C_{50}$

$0 = C_0 - C_1 + C_2 - C_3 + \dots + C_{50}$ --- (Equation 2)

4. Solve for Odd Powers

Subtract Equation 2 from Equation 1:

$(C_0 + C_1 + C_2 + \dots) - (C_0 - C_1 + C_2 - \dots) = 2^{50} - 0$

$2(C_1 + C_3 + C_5 + \dots + C_{49}) = 2^{50}$

Divide by $2$ :

$C_1 + C_3 + C_5 + \dots + C_{49} = \dfrac{2^{50}}{2}$

$C_1 + C_3 + C_5 + \dots + C_{49} = 2^{49}$

Final Answer:

The sum of coefficients of odd powers of $x$ is $2^{49}$ .

Choose the correct answer:

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