Question

How do I find the ${n^{th}}$ term of a binomial expansion?

Answer 1

Hint: First write down the binomial expression and then write its expansion. The expansion should at least contain $2 - 3$ terms from the beginning and $2 - 3$ terms from the end. Check out the pattern of the progressing terms and then write the general formula for the ${n^{th}}$ term for the binomial expansion.

Complete step-by-step answer:
Let’s write the ${n^{th}}$ term for the binomial expression, ${(a + b)^n}$
Here, $a,b\;$ are real numbers and $n$ is a positive integer.
${(a + b)^n}$ when expanded we get,
$\Rightarrow {(a + b)^n}{ = ^n}{C_0}{ + ^n}{C_1}{a^{n - 1}}{b^1}{ + ^n}{C_2}{a^{n - 2}}{b^2} + ........{ + ^n}{C_n}{b^n}$
Where $^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}$
$r$ is any number in the range,
As we can see that every term follows a pattern which is,
The power of $a$ keeps on consecutively decreasing, whereas that of $b$ increases progressively.
So, collectively we can write the changes as,
Considering $r$ is any ${n^{th}}$ term.
$\Rightarrow {T_{r + 1}}{ = ^n}{C_r}{a^{n - r}}{b^r}$
We can see that $r$ in the $^n{C_r}$ keeps on increasing, the power of $b$ will be the same as $r$
And the power of $a$ will be $n - r$ .
Substitute any value in place of $r$ to cross-check if we are getting the same term as in the sequence.

$\therefore$ The ${n^{th}}$ term of binomial expansion is ${T_{r + 1}}{ = ^n}{C_r}{a^{n - r}}{b^r}$

Additional information: A binomial is a mathematical expression that contains two terms that are together by any of the operations either addition or subtraction. The coefficients of the binomial expansion follow a determined pattern which is known as Pascal’s Triangle. Considering a binomial expansion, any term in it, the sum of all the exponents of $a$ and $b$ is always equal to $n$ . Here, $n$ is the power of the binomial expression.

Note:
Whenever a binomial expression is given, always write its expansion. If the power of the binomial expression is $n$ then the total number of all the terms in the expansion is $(n + 1)$ . One must not forget about this, if not it would go wrong while finding the last term.