Binomial Theorem Formula

About Binomial Theorem Formula

In order to expand any binomial power into a series, the binomial theorem formula is needed. The formula for the binomial theorem is (a+b)ⁿ = ⁿC_ra^n-rb^r, Here n is a positive integer, a, b belongs to real numbers, and 0 < r ≤ n. This formula can be used to expand binomial expressions like (x + a)¹⁰, (2x + 5)³, (x- (1/x))⁴, and so on.The formula for the binomial theorem aids in the expansion of a binomial raised to a particular power.

If x and y are both real numbers, then the binomial theorem holds for all n ∈ N,

(x + y)ⁿ = ⁿc₀xⁿy⁰ + ⁿc₁x^n-1y¹ + ⁿc₂x^n-2y²-------+ⁿc_n-1x¹y^n-1 + ⁿc_nx⁰y⁰

⇒ (x + y)ⁿ = ⁿC_k x^{n - k}y ^k

where, ⁿC_r = n! / [r! (n - r)!]

Binomial Theorem Expansion Proof

Let x, a, n ∈ N. Let us use the idea of mathematical induction to show the binomial theorem formula. It is sufficient to demonstrate for n = 1, n = 2, and n = 3, for n = k ≥ 2, and for n = k+ 1.

obviously (x +y)¹= x +y and

1. (x +y)²= (x + y) (x +y)
2. = x²+ xy + xy + y²
3. = x²+ 2xy + y²

As a result, the result holds for both n = 1 and n = 2. Make k a positive number. Let’s prove that the result is true for k ≥ 2.

By assuming (x + y)ⁿ= ⁿC_rx^n-ry^r,

(x + y)^k= ^kC_rx^k-ry^r

⇒ (x+y)^k=^kC₀x^ky⁰+^kC₁x^k-1y¹+^kC₂x^k-2y²+ ... +^kC_rx^k-ry^r+....+^kC_kx^k-ky^k

Hence, this result is true for n = k ≥ 2.

Now by considering the expansion also for n = k + 1.

(x + y)^k+1= (x + y) (x + y)^k

= (x + y) (x^k+^kC₁x^k-1y¹+^kC₂x^k-2y²+ ... +^kC_rx^k-ry^r+....+ y^k)

= x^k+1+^k+1C₁x^ky +^k+1C₂x^k-1y2 + ... +^k+1C_rx^k-ry^r + …... +^k+1C_kxy^k + y^k+1[BecauseⁿC_r+ⁿC_r-1=ⁿ⁺¹C_r]

Thus the result is true for n = k+1. By mathematical induction, this result is true for all positive integers 'n'. Hence proved.

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