Binomial Theorem — Part 2: General term, middle terms and applications: visual lesson, CBSE Class 11 Mathematics

Lesson

Binomial theorem, Part 2: Finding one term at a time

The expansion of $(x + 2)^{6}$ has seven terms. How can we find just the fourth, without writing out the rest?

In this lesson you will

Write the general term of the expansion
Find a particular term and the term independent of $x$
Find the middle term or terms
Use equal binomial coefficients to find an unknown power

Seven terms for the sixth power.
Term	$r$	$\binom{6}{r}x^{6-r}\,2^{r}$	Simplified
$T_{1}$	$0$	$\binom{6}{0}x^{6}2^{0}$	$x^{6}$
$T_{2}$	$1$	$\binom{6}{1}x^{5}2^{1}$	$12x^{5}$
$T_{3}$	$2$	$\binom{6}{2}x^{4}2^{2}$	$60x^{4}$
$T_{4}$	$3$	$\binom{6}{3}x^{3}2^{3}$	$160x^{3}$
$T_{5}$	$4$	$\binom{6}{4}x^{2}2^{4}$	$240x^{2}$
$T_{6}$	$5$	$\binom{6}{5}x^{1}2^{5}$	$192x$
$T_{7}$	$6$	$\binom{6}{6}x^{0}2^{6}$	$64$

Concept

What you will be able to do

Write the general term $T_{r+1}$
Find a particular term and the term independent of $x$
Find the middle term(s)
Use equal binomial coefficients to find $n$

Concept

The general term

In $(a + b)^{n}$ the $(r+1)$ th term is $T_{r+1} = \dbinom{n}{r}a^{n-r}b^{r}$ , for $r = 0, 1, \ldots, n$ . The term number is always one more than $r$ .

T_{r+1} = \binom{n}{r}a^{n-r}b^{r}

The $(r+1)$th term has $b^{r}$ and the coefficient $\binom{n}{r}$.
Term number	$r$	$T_{r+1} = \binom{7}{r}a^{7-r}b^{r}$	Coefficient
$T_{1}$	$0$	$\binom{7}{0}a^{7}b^{0}$	$1$
$T_{2}$	$1$	$\binom{7}{1}a^{6}b^{1}$	$7$
$T_{3}$	$2$	$\binom{7}{2}a^{5}b^{2}$	$21$
$T_{4}$	$3$	$\binom{7}{3}a^{4}b^{3}$	$35$
$T_{5}$	$4$	$\binom{7}{4}a^{3}b^{4}$	$35$
$T_{6}$	$5$	$\binom{7}{5}a^{2}b^{5}$	$21$
$T_{7}$	$6$	$\binom{7}{6}a^{1}b^{6}$	$7$
$T_{8}$	$7$	$\binom{7}{7}a^{0}b^{7}$	$1$

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