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Methods of Induction and Binomial Theorem

Maharashtra Board · Class 11 · Mathematics & Statistics

Quick revision notes for Methods of Induction and Binomial Theorem — Maharashtra Board Class 11 Mathematics & Statistics. Key concepts, formulas, and definitions for last-minute revision.

45 questions20 flashcards5 concepts

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Key Topics to Revise

1

Mathematical Induction - Fundamental Principles

  • Mathematical Induction has FOUR essential steps that must ALL be completed
  • Step 1 (Foundation): Prove P(1) is true - this is your starting point
  • Step 2 (Assumption): Assume P(k) is true for some positive integer k
2

Binomial Theorem for Positive Integral Index

  • Binomial Theorem: (a+b)ⁿ = Σ(r=0 to n) ⁿCᵣ aⁿ⁻ʳ bʳ
  • Expansion has exactly (n+1) terms
  • First term is aⁿ, last term is bⁿ
3

General Term and Middle Terms

  • General term: Tᵣ₊₁ = ⁿCᵣaⁿ⁻ʳbʳ (this is the (r+1)th term)
  • For coefficient of xᵐ: Set the power of x equal to m and solve for r
  • For constant term: Set the power of x equal to zero
4

Binomial Theorem for Negative and Fractional Index

  • For |x| < 1 and any real n: (1+x)ⁿ = 1 + nx + n(n-1)/2!·x² + n(n-1)(n-2)/3!·x³ + ...
  • This gives INFINITE terms (unlike positive integer case)
  • General term: Tᵣ₊₁ = [n(n-1)(n-2)...(n-r+1)/r!]xʳ

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Key Concepts

A four(a+b)^n = ∑(r=0 to n) nCrThe (r+1)th term in (a+b)^nIf n is evenFor |x| < 1 and any

Frequently Asked Questions

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Sources & Official References

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