What is the general term of the binomial expansion?

T_(r+1) = ⁿCᵣ × a^(n-r) × b^r, where r = 0, 1, 2, ..., n. This formula gives the (r+1)th term directly.

How do you find the middle term of a binomial expansion?

If n is even, there is one middle term: T_(n/2 + 1). If n is odd, there are two middle terms: T_((n+1)/2) and T_((n+3)/2).

What is the sum of binomial coefficients?

Put a = b = 1 in (a+b)ⁿ: (1+1)ⁿ = 2ⁿ. So ⁿC₀ + ⁿC₁ + ... + ⁿCₙ = 2ⁿ.

Binomial Expansion

Binomial Theorem • Class 11 Mathematics • NCERT • CBSE

(a+b)ⁿ = Σ ⁿCᵣ aⁿ⁻ʳ bʳ. General term: T_(r+1) = ⁿCᵣ aⁿ⁻ʳ bʳ. Middle term: if n even → (n/2+1)th term; if n odd → two middle terms. Number of terms = n+1.

Key Formulas

(a+b)ⁿ = Σ ⁿCᵣ aⁿ⁻ʳ bʳ
T_(r+1) = ⁿCᵣ aⁿ⁻ʳ bʳ (General term)
Sum of coefficients = 2ⁿ
Sum of alternate coefficients = 2ⁿ⁻¹ each

Frequently Asked Questions

What is the general term of the binomial expansion?: T_(r+1) = ⁿCᵣ × a^(n-r) × b^r, where r = 0, 1, 2, ..., n. This formula gives the (r+1)th term directly.
How do you find the middle term of a binomial expansion?: If n is even, there is one middle term: T_(n/2 + 1). If n is odd, there are two middle terms: T_((n+1)/2) and T_((n+3)/2).
What is the sum of binomial coefficients?: Put a = b = 1 in (a+b)ⁿ: (1+1)ⁿ = 2ⁿ. So ⁿC₀ + ⁿC₁ + ... + ⁿCₙ = 2ⁿ.

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