Question 8 of 20intermediate🔧 ApplyNumerical4 marks

Using binomial theorem, evaluate each of the following: (101)^4

Correct Answer

104060401

Exercise: EXERCISE 7.1 | Q: 8 | (Page 8)
For More Understanding

Explanation

Following the approach in Example 2 of the textbook, we express 101 as the sum of two numbers whose powers are easier to calculate. We write 101 = 100 + 1, then apply the Binomial Theorem to expand (100 + 1)^4. The expansion uses the formula (a + b)^n = ⁿC₀aⁿ + ⁿC₁aⁿ⁻¹b + ⁿC₂aⁿ⁻²b² + ... + ⁿCₙbⁿ with the binomial coefficients for n=4 being 1, 4, 6, 4, 1.

Solution Steps

  1. Step 1: Express 101 as sum of convenient numbers: 101 = 100 + 1

  2. Step 2: Apply Binomial Theorem: (101)⁴ = (100 + 1)⁴ = ⁴C₀(100)⁴ + ⁴C₁(100)³(1) + ⁴C₂(100)²(1)² + ⁴C₃(100)(1)³ + ⁴C₄(1)⁴

  3. Step 3: Substitute binomial coefficients: = 1×(100)⁴ + 4×(100)³×1 + 6×(100)²×1 + 4×1004 \times 100×1 + 1×11 \times 1

  4. Step 4: Calculate each term: = 100000000 + 4000000 + 60000 + 400 + 1

  5. Step 5: Add all terms: = 104060401

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