Binomial-theorem Formula Sheet

The binomial series expansion expresses the power of a binomial (a two-term expression) as an infinite sum of terms involving binomial coefficients, following the formula
$(a+b)^n = \displaystyle\sum_{k=1}^n \binom{n}{k} a^{n-k}b^k$

Neetesh Kumar | May 08, 2024 $\space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space$ Share this Page on:

• 1. Binomial Expansion Formula
• 2. Important Terms
• 3. Some results on Binomial Coefficients
• 4. Greatest Coefficients and Term in the expansion of (x+a)$^n$
• 5. Binomial Theorem for Negative and fractional Indices
• 6. Exponential Series
• 7. Logarithmic Series

1. Binomial Expansion Formula :

$(x + y)^n =n_{C_0}x^n + n_{C_1}x^{n-1}y + n_{C_2}x^{n-2}y^2 + .... + n_{C_r}x^{n-r}y^r + n_{C_n}y^n = \displaystyle\sum_{r=0}^n n_{C_r}x^{n-r}y^r$ where n $\in$ N.

2. Important Terms:

$\bold{General \space Terms:}$ The general term or the (r+1)$^{th}$ term in the expansion of (x+y)$^{n}$ is given by
$T_{r+1} = n_{C_r}x^{n-r}y^r$

$\bold{Middle \space Term:}$ The middle terms in the expansion of (x+y)$^n$ are:

• If n is even, there is only one middle term which is given by
  $T_{\frac{n+2}{2}} = n_{C_\frac{n}{2}}x^{\frac{n}{2}}y^{\frac{n}{2}}$
  

• If n is odd, there are two middle terms, which are $T_{\frac{n+1}{2}}$ and $T_{[\frac{n+1}{2}] + 1}$

$\bold{Term \space Independent \space of \space x:}$ Term independent of x contains no x;
Hence, find the value of r for which the exponent of x is zero.

3. Some results on Binomial Coefficients:

$\bold{(a)}$ $n_{C_x} = n_{C_y} \Rightarrow x = y$ or $x+y=n$
$\bold{(b)}$ $n_{C_{r-1}} + n_{C_r} = {n+1}_{C_r}$
$\bold{(c)}$ $C_0 + C_1 + C_2 + .... + C_n = 2^n,$ where $C_r = n_{C_r}$
$\bold{(d)}$ $C_0 + C_2 + C_4 + .... = C_1 + C_3 + C_5 + .... = 2^{n-1},$ where $C_r = n_{C_r}$
$\bold{(c)}$ $C_0^2 + C_1^2 + C_2^2 + .... + C_n^2 = 2n_{C_n} = \frac{(2n)!}{n!n!},$ where $C_r = n_{C_r}$

4. Greatest Coefficients and Term in the expansion of (x+a)$^n$:

$\bold{(a)}$ If n is even, greatest binomial coefficients is $n_{C_\frac{n}{2}}$
If n is odd, greatest binomial coefficients is $n_{C_\frac{n-1}{2}}$ or $n_{C_\frac{n+1}{2}}$
$\bold{(b)}$ For Greatest Term:
$Greatest \space term = \begin{cases} T_p \space and \space T_{p+1} &\text{if } \frac{n+1}{|\frac{x}{a}|+1} \space is \space an \space integer \space equal \space to \space p \\ T_{q+1} &\text{if } \frac{n+1}{|\frac{x}{a}|+1} \space is \space non \space integer \space and \in (q, q+1), q \in I \end{cases}$

5. Binomial Theorem for Negative and fractional Indices:

If n $\in$ R, then $(1+x)^n = 1 + nx + \frac{n(n-1)}{2!}x^2 + \frac{n(n-1)(n-2)}{3!}x^3 + .... \infty$ provided | x | < 1
Note:
$\bold{(i)}$ $(1-x)^{-1} = 1 + x + x^2 + x^3 + .... \infty$
$\bold{(ii)}$ $(1+x)^{-1} = 1 - x + x^2 - x^3 + .... \infty$
$\bold{(iii)}$ $(1-x)^{-2} = 1 + 2x + 3x^2 + 4x^3 + .... \infty$
$\bold{(iv)}$ $(1+x)^{-2} = 1 - 2x + 3x^2 - 4x^3 + .... \infty$

6. Exponential Series:

$\bold{(a)}$ $e^x = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!}+.... \infty;$ where x may be any real or complex number and e = $\lim\limits_{x\to \infty}(1+\frac{1}{n})^n$

$\bold{(b)}$ $a^x = 1 + \frac{x}{1!}lna + \frac{x^2}{2!}ln^2a + \frac{x^2}{3!}ln^3a + .... \infty,$ where a > 0

7. Logarithmic Series:

$\bold{(a)}$ $ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + .... \infty$ where -1 < x $\le$ 1
$\bold{(b)}$ $ln(1-x) = -x - \frac{x^2}{2} - \frac{x^3}{3} - \frac{x^4}{4} - .... \infty$ where -1 $\le$ x < 1
$\bold{(c)}$ $ln(\frac{1+x}{1-x}) = 2(x + \frac{x^3}{3} + \frac{x^5}{5} + .... \infty)$ where | x | < 1

$\color{red} \bold{Related \space Pages:}$
Binomial Expansion Calculators
Vector operation Calculators
Vector Formula sheet