General Term
The general term is the single most useful formula in the chapter: T(r+1) = nCr · a^(n−r) · b^r. It is the (r+1)th term, so r runs from 0 (first term) to n (last term) — the +1 trips up many students, so anchor it: r = 0 gives the first term, not r = 1. To find a SPECIFIC term, you do not expand anything — you set up the general term and solve for r. For "find the term containing x^k" type questions, collect all powers of x in T(r+1) into a single exponent of x, set that exponent equal to k, and solve for r; then plug r back to read the coefficient. This is the CAT-fast method and it works identically whether a and b are plain numbers, powers of x, or fractions like x/2 and 3/x². Always simplify the combined power of x carefully — that exponent equation is where the marks are won or lost.
✅ 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) |