How to Find Middle Terms in a Binomial

Quick answer: The expansion of (a + b)ⁿ contains n + 1 formal terms. If n is even there is one middle term, Tₙ/₂₊₁. If n is odd there are two, T₍ₙ₊₁₎/₂ and T₍ₙ₊₃₎/₂.

Worked example

Find the middle term of (x + 1)⁶ and the middle terms of (x + 1)⁵.

1. For n = 6, seven terms means the fourth is central.
2. T₄ = 6C₃x³ = 20x³.
3. For n = 5, six terms means the third and fourth are central.
4. They are 5C₂x³ = 10x³ and 5C₃x² = 10x².

Check the result

The exponents in the first expansion are 6, 5, 4, 3, 2, 1, 0; x³ sits in the centre.

Common mistake to avoid

Count terms using n + 1, not n. The binomial coefficient index is one less than the term number.

Try it yourself

1. Find the middle term of (a + b)⁴. 2. Find both middle terms of (x + 2)³.

Answers and reasoning

1. T₃ = 6a²b². 2. T₂ = 6x² and T₃ = 12x.

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