Binomial Theorem: General Term and Middle Term | NIMCET

Quick answer: For a positive integer n, $(x + a)^n = \sum_{r=0}^{n}{}^nC_r\,x^{n - r}a^r$ , with $n + 1$ terms. The general term is $T_{r + 1} = {}^nC_r\,x^{n - r}a^r$ . If n is even, the single middle term is $T_{\frac{n}{2} + 1}$ ; if n is odd, there are two, $T_{\frac{n + 1}{2}}$ and $T_{\frac{n + 3}{2}}$ . The sum of all coefficients is $2^n$ .

The binomial theorem is part of the revised NIMCET algebra syllabus, as listed in the NIMCET syllabus update. The coefficients are the $^nC_r$ values from combinations.

The expansion

$(x + a)^n = {}^nC_0x^n + {}^nC_1x^{n - 1}a + {}^nC_2x^{n - 2}a^2 + \cdots + {}^nC_na^n$

Fact	Value
Number of terms	$n + 1$
General term	$T_{r + 1} = {}^nC_r\,x^{n - r}a^r$
Powers	Power of x falls from n to 0; power of a rises from 0 to n
$(x - a)^n$	Signs alternate: $T_{r + 1} = (-1)^r\,{}^nC_r\,x^{n - r}a^r$
$r$ th term from the end	$(n - r + 2)$ th term from the start

Middle terms

n	Middle term(s)
Even	$T_{\frac{n}{2} + 1}$
Odd	$T_{\frac{n + 1}{2}}$ and $T_{\frac{n + 3}{2}}$

The middle term has the greatest binomial coefficient.

Term independent of x

For an expansion like $\left(x^p + \frac{1}{x^q}\right)^n$ , write the power of x in $T_{r + 1}$ , set it equal to 0, and solve for r. If r is not a whole number from 0 to n, there is no such term.

Properties of binomial coefficients

Write $C_r$ for $^nC_r$ .

Property	Result
Sum of all coefficients	$C_0 + C_1 + \cdots + C_n = 2^n$
Alternating sum	$C_0 - C_1 + C_2 - \cdots = 0$
Even-position sum	$C_0 + C_2 + C_4 + \cdots = 2^{n - 1}$
Odd-position sum	$C_1 + C_3 + C_5 + \cdots = 2^{n - 1}$
Weighted sum	$C_1 + 2C_2 + 3C_3 + \cdots + nC_n = n\,2^{n - 1}$
Sum of squares	$C_0^2 + C_1^2 + \cdots + C_n^2 = {}^{2n}C_n$

Sum of coefficients of any expansion: put every variable equal to 1. For example, the coefficients of $(2x - 3y)^5$ add to $(2 - 3)^5 = -1$ .

Worked examples

Example 1: a specific term

Find the 4th term of $(2x + 3)^6$ .

$T_4 = {}^6C_3(2x)^3(3)^3 = 20 \cdot 8x^3 \cdot 27 = 4320x^3$

Example 2: term independent of x

Find the term independent of x in $\left(x^2 + \frac{1}{x}\right)^9$ .

Step 1: $T_{r + 1} = {}^9C_r\,x^{18 - 2r}x^{-r} = {}^9C_r\,x^{18 - 3r}$ .

Step 2: $18 - 3r = 0$ , so $r = 6$ .

Answer: $T_7 = {}^9C_6 = 84$ .

Example 3: middle term

Find the middle term of $\left(x - \frac{1}{x}\right)^{10}$ .

n is 10, so the middle term is $T_6$ :

$T_6 = {}^{10}C_5\,x^5\left(-\frac{1}{x}\right)^5 = -252$

Example 4: coefficient of a power

Find the coefficient of $x^5$ in $(1 + x)^8$ .

$^8C_5 = 56$ .

Example 5: remainder

Find the remainder when $7^{100}$ is divided by 48.

Step 1: $7^{100} = (49)^{50} = (1 + 48)^{50}$ .

Step 2: Every term after the first contains 48, so the remainder is 1.

Summary table

Idea	Key result
General term	$^nC_r\,x^{n - r}a^r$
Number of terms	$n + 1$
Middle term (n even)	$T_{\frac{n}{2} + 1}$
Sum of coefficients	$2^n$ (put x = a = 1)
$\sum C_r^2$	$^{2n}C_n$

Exam tips

Remember that $T_{r + 1}$ , not $T_r$ , uses $^nC_r$ .
For the term independent of x, write the power of x as a function of r first.
For remainder questions, rewrite the base as (multiple of the divisor) ± 1.

See all topics in the NIMCET Algebra chapter.

Common mistakes

Using $^nC_r$ for the rth term instead of the $(r + 1)$ th.
Forgetting the negative sign in expansions of $(x - a)^n$ .
Treating the binomial coefficient as the full coefficient when a or x carries a number.
Counting n terms instead of $n + 1$ .

Frequently Asked Questions

What is the general term of (x + a)ⁿ?

T₍ᵣ₊₁₎ = ⁿCᵣ xⁿ⁻ʳ aʳ.

How many terms are in the expansion of (x + a)ⁿ?

There are n + 1 terms.

What is the middle term when n is even?

It is the (n/2 + 1)th term.

What is the sum of binomial coefficients?

ⁿC₀ + ⁿC₁ + … + ⁿCₙ = 2ⁿ.

How do you find the term independent of x?

Write the power of x in the general term, set it to zero, and solve for r.