Middle Term
The middle term depends only on whether n is even or odd. If n is EVEN there is one middle term, the ((n/2)+1)th term, i.e. T(n/2 + 1) with r = n/2. If n is ODD there are TWO middle terms, the ((n+1)/2)th and ((n+3)/2)th terms, with r = (n−1)/2 and r = (n+1)/2. A quick way to remember it: (n+1) terms split evenly around a single centre when (n+1) is odd (n even), and around a pair when (n+1) is even (n odd). Once you know r, the middle term is just the general term evaluated there: T = nCr · a^(n−r) · b^r. CAT loves to combine this with "term independent of x" — sometimes the middle term IS the constant term, but not always, so compute the power of x rather than assuming. Get the parity of n right first; everything else follows from the general term.
✅ Solved examples
✏️ Practice — try these, take hints as needed
📝 Topic test — 8 questions
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Formula Reference Sheet
Expansion & general term
| Binomial expansion | (a+b)ⁿ = Σ nCr · a^(n−r) · b^r, r = 0…n |
|---|---|
| General term | T(r+1) = nCr · a^(n−r) · b^r |
| Number of terms | (a+b)ⁿ has (n+1) terms |
| Binomial coefficient | nCr = n! / [r!(n−r)!] |
| Symmetry of coefficients | nCr = nC(n−r) |
Middle term, sums & special cases
| Middle term (n even) | single term T(n/2 + 1) |
|---|---|
| Middle terms (n odd) | two terms T((n+1)/2) and T((n+3)/2) |
| Sum of all coefficients | put each variable = 1 ⇒ (sum of bases)ⁿ |
| Sum of binomial coefficients | nC0 + nC1 + … + nCn = 2ⁿ |
| Greatest coefficient | the middle coefficient: nC(n/2) (n even) |