in the expansion of (1+x)^50, the sum of the coefficients of odd powers of x is:
Answers
Sum of the coefficient of odd powers of x in (1 + x)⁵⁰ = 2⁴⁹
Step-by-step explanation:
(1 + x)⁵⁰ = 1 + ⁵⁰C₁x¹ + ⁵⁰C₂x² + ⁵⁰C₃x³ +.......................+ ⁵⁰C₄₉x⁴⁹ + ⁵⁰C₅₀x⁵⁰
the sum of the coefficients of odd powers of x is:
= ⁵⁰C₁ + ⁵⁰C₃ + .............................. + ⁵⁰C₄₉
(1 - x)⁵⁰ = 1 + ⁵⁰C₁(-x)¹ + ⁵⁰C₂(-x)² + ⁵⁰C₃(-x)³ +................+ ⁵⁰C₄₉(-x)⁴⁹ + ⁵⁰C₅₀(-x)⁵⁰
= 1 - ⁵⁰C₁x¹ + ⁵⁰C₂x² - ⁵⁰C₃x³ +.......................- ⁵⁰C₄₉x⁴⁹ + ⁵⁰C₅₀x⁵⁰
(1 + x)⁵⁰ - (1 - x)⁵⁰ = 2(⁵⁰C₁x¹ + ⁵⁰C₃x³ +.................................+⁵⁰C₄₉x⁴⁹)
putting x = 1
=> 2⁵⁰ -0 = 2 ( ⁵⁰C₁ + ⁵⁰C₃ + .............................. + ⁵⁰C₄₉)
=> 2⁵⁰ = 2 ( ⁵⁰C₁ + ⁵⁰C₃ + .............................. + ⁵⁰C₄₉)
=> ⁵⁰C₁ + ⁵⁰C₃ + .............................. + ⁵⁰C₄₉ = 2⁵⁰/2
=> ⁵⁰C₁ + ⁵⁰C₃ + .............................. + ⁵⁰C₄₉ = 2⁴⁹
Sum of the coefficient of odd powers of x in (1 + x)⁵⁰ = 2⁴⁹
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