The General Term in the Expansion

Deriving the General-Term Formula

Writing out the binomial theorem in full,

notice that the first term ($^{n}C_0a^n$) is the $1$st term, the second term ($^{n}C_1a^{n-1}b$) is the $2$nd term, and so on -- the term containing $^{n}C_r$ is always the $(r+1)$th term of the expansion, because the count of terms starts from $r=0$ while the position count starts from $1$.

Writing $T_{r+1}$ for the $(r+1)$th term, the general term of the expansion of $(a+b)^n$ is

Using the General Term

The general term lets us answer two common questions without expanding the whole binomial.

Finding a specific term. To find the $p$th term of an expansion, set $r+1=p$, i.e. $r=p-1$, and substitute into $T_{r+1}$. For example, the $4$th term of $(a+b)^n$ is $T_4=T_{3+1}$, so $r=3$, giving $T_4={}^{n}C_3a^{n-3}b^3$.

Finding the coefficient of a given power. To find the coefficient of a particular power of one of the variables (say $x^m$ in an expansion involving $x$), first write the general term $T_{r+1}$ for the given binomial, simplify the power of $x$ that it carries to a single expression in $r$, set that expression equal to $m$, solve for $r$, and substitute back to get the coefficient. If a binomial has the form $\left(x^p+\dfrac{c}{x^q}\right)^n$, the general term is

$T_{r+1}={}^{n}C_r\,(x^p)^{n-r}\left(\frac{c}{x^q}\right)^r = {}^{n}C_r\,c^r\,x^{p(n-r)-qr},$ …