Prove that the sum of the coefficient of the odd powers of x the expansion of (1 + x + x2 + x3 + x4)n−1, when n is a prime number other than 5, is divisible by n.

Engineering

Mathematics

Introduction to Binomial Theorem

Application of Binomial Theorem and Summation Of Series

Exponential and Basic Maths questions

Question

Prove that the sum of the coefficient of the odd powers of x the expansion of (1 + x + x² + x³ + x⁴)ⁿ⁻¹, when n is a prime number other than 5, is divisible by n.

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Given,
n is a prime number other than 5. Also given equation,

(1 + x + x² + x³ + x⁴)ⁿ^–¹ = 1 + C₁x + C₂x² + C₃x³ + C₄x⁴+.....(1)

(1 – x + x² – x³ + x⁴)ⁿ^–¹ = 1 – C₁x + C₂x² – C₃x³ + C₄x⁴+.....(2)

Substituting x = 1 then,
subracting (1) and (2)

⇒ 5ⁿ^–¹ – 1 = 2(C₁ + C₃ + C₅.....)

$\Rightarrow C_{1} + C_{3} + C_{5} = \frac{5^{n - 1} - 1}{2}$

using fermats theorem,

If N is prime to p the N^p^–¹ – 1 is divisible by P
⇒ 5ⁿ^–¹ – 1 is divisible by n

where, n is prime other than 5.

Hence proved.