Hard

The General Term of a Binomial Expansion

Simple Explanation

The general term formula, T(r+1) = ⁿCᵣ aⁿ⁻ʳ bʳ, lets you jump directly to any single term of a binomial expansion — like the 5th term, or the term containing a particular power of x — without expanding every earlier term first.

Why Do We Need It?

For a high power n, expanding the entire binomial just to reach one specific term wastes a huge amount of effort — the general term formula gets there in one step.

Formula

The General Term of a Binomial Expansion

T(r+1) = ⁿCᵣ · aⁿ⁻ʳ · bʳ

Gives the formula for the (r+1)th term of the expansion of (a+b)ⁿ directly, without writing out every earlier term first.

T(r+1): — the (r+1)th term of the expansion (the "+1" exists because the first term corresponds to r=0)
n, r: — as in the binomial theorem — n is the power, r is the term index

When to use it: Whenever you need one specific term of a binomial expansion (e.g. the 5th term, or the term containing x⁴) without expanding the whole thing.

Worked Example

Find a specific term directly

Find the 4th term in the expansion of (x + 2)⁷.

Why Does This Work?

The general term is just the binomial theorem's summation formula with one specific value of r plugged in — since every term of the full sum already has this exact form, isolating one value of r extracts exactly that one term, correctly, without needing the rest.

Real-Life Example

Finding a specific data-fitting coefficient

An engineer needs only one specific coefficient from a large binomial-based series expansion used in a signal-processing model.

Using the general term formula for that one r-value avoids computing the entire (potentially huge) expansion just to reach the needed term.

Practice

What is the 3rd term in the expansion of (x + 1)⁵?

Hard

Common mistake

Using r = (term number) instead of r = (term number) − 1 — the Tth term corresponds to r = T−1, since the very first term (T1) always corresponds to r=0.

Quick Review

T(r+1) = ⁿCᵣ aⁿ⁻ʳ bʳ.
For the Tth term, use r = T − 1.
Lets you find one specific term without expanding the whole binomial.