Binomial Theorem General Term

Quick answer: In (a + b)ⁿ for a nonnegative integer n, Tᵣ₊₁ = nCᵣ aⁿ⁻ʳbʳ. The (r + 1)th term uses r, so the fourth term has r = 3.

Worked example

Find the fourth term of (2x + 3)⁵.

1. Set n = 5, a = 2x, b = 3 and r = 3.
2. T₄ = 5C₃(2x)²3³.
3. 5C₃ = 10, (2x)² = 4x² and 3³ = 27.
4. Multiply to get 1080x².

Check the result

The power of x drops from 5 to 2 by the fourth term: 5, 4, 3, 2.

Common mistake to avoid

Do not use r = 4 for the fourth term. Raise the whole bracket 2x to the power, not x alone.

Try it yourself

1. Find T₃ of (x + 2)⁴. 2. Find T₂ of (3x − 1)³.

Answers and reasoning

1. 4C₂x²2² = 24x². 2. 3C₁(3x)²(−1) = −27x².

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