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Mathematics

Statement and proof of the binomial theorem for positive integral indices

In this Class 11 Mathematics topic from the Binomial Theorem chapter, students learn the statement of the binomial theorem for a positive integral index and understand how it is proved. The theorem expresses the expansion of (a + b)^n as a finite sum involving binomial coefficients. Students examine the pattern of terms, powers of a and b, and the role of combinations in determining coefficients, building the foundation for finding specific terms and using binomial expansions accurately.

Medium · Level 1 · 1 questions

Practice questions

01The identity Σ from r = 0 to n of (-1)^r ⁿCᵣ = 0 is connected with which substitution?

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Answer and explanation

Correct answer: C. Put x = -1 in (1 + x)ⁿ

The governing concept is the binomial theorem, (1+x)ⁿ = Σ from r=0 to n of ⁿCᵣxʳ. The required identity contains the factor (−1)^r, so x must be replaced by −1; this changes xʳ into (−1)^r. The left side then becomes (1−1)ⁿ=0ⁿ=0 for positive n, while the expansion becomes Σ ⁿCᵣ(−1)^r. Therefore option C is correct. If x=1 is used, the result is the sum of all positive coefficients, 2ⁿ. If x=0 is used, all terms except the constant term vanish. Taking x=n creates no alternating sign pattern and cannot produce the stated identity.

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Statement and proof of the binomial theorem for positive integral indices

Practice questions

01The identity Σ from r = 0 to n of (-1)^r ⁿCᵣ = 0 is connected with which substitution?Σ, जहाँ r = 0 से n तक (-1)^r ⁿCᵣ का मान 0 है, किस प्रतिस्थापन से प्राप्त होता है?

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01The identity Σ from r = 0 to n of (-1)^r ⁿCᵣ = 0 is connected with which substitution?