Binomial Theorem

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A statement is a definite assertion that can be either true or false, but not both. Example: 'x + 5 = 7 ⟹ x = 2' is a statement (true), while 'x + 5 = 7' is not a statement because its truth depends o

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Let P(n) be a statement involving natural number n. P(n) is true for all n ∈ ℕ if: (i) P(1) is true (Base Step) (ii) P(k) being true implies P(k+1) is true for any k ∈ ℕ (Induction Step) Both conditio

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Step 1 (Base): P(1): 1 = 1(2)/2 = 1 ✓ Step 2 (Assumption): Assume P(k): 1+2+...+k = k(k+1)/2 Step 3 (Induction): P(k+1): 1+2+...+k+(k+1) = (1+2+...+k)+(k+1) = k(k+1)/2 + (k+1) = (k+1)[k/2 + 1] = (k+1)

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A binomial is an algebraic expression with exactly two terms connected by '+' or '-'. Examples: 1. (x + y) 2. (2a - 3b) 3. (x² + 1/x) Note: The terms can be any algebraic expressions, not just simple

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(x + y)ⁿ = ⁿC₀xⁿ + ⁿC₁xⁿ⁻¹y + ⁿC₂xⁿ⁻²y² + ... + ⁿCᵣxⁿ⁻ʳyʳ + ... + ⁿCₙyⁿ Where ⁿCᵣ = n!/(r!(n-r)!) are binomial coefficients. Alternatively: (x + y)ⁿ = Σ(r=0 to n) ⁿCᵣxⁿ⁻ʳyʳ

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(x + 2y)⁴ = ⁴C₀x⁴ + ⁴C₁x³(2y) + ⁴C₂x²(2y)² + ⁴C₃x(2y)³ + ⁴C₄(2y)⁴ = 1·x⁴ + 4·x³·2y + 6·x²·4y² + 4·x·8y³ + 1·16y⁴ = x⁴ + 8x³y + 24x²y² + 32xy³ + 16y⁴

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The general term is Tᵣ₊₁ = ⁿCᵣxⁿ⁻ʳyʳ Where: - r ranges from 0 to n - Tᵣ₊₁ represents the (r+1)th term - The first term (r=0) is ⁿC₀xⁿ - The last term (r=n) is ⁿCₙyⁿ

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For the 5th term: T₅ = T₄₊₁, so r = 4 T₅ = ⁸C₄(2x)⁸⁻⁴(-1/x)⁴ = ⁸C₄(2x)⁴(1/x⁴) = 70 × 16x⁴ × (1/x⁴) = 70 × 16 = 1120 Note: ⁸C₄ = 8!/(4!4!) = 70