# General Term in Binomial Expansion

Here you will learn formula to find the general term in binomial expansion with examples.

Let’s begin –

## General Term in Binomial Expansion

We have,

$$(x + a)^n$$ = $$^{n}C_0 x^n a^0$$ + $$^{n}C_1 x^{n – 1} a^1$$ + …………… + $$^{n}C_r x^{n – r} a^r$$ + …………… + $$^{n}C_n x^0 a^n$$

We find that : The first term = $$^{n}C_0 x^n a^0$$

The second term = $$^{n}C_1 x^{n – 1} a^1$$

The third term = $$^{n}C_2 x^{n – 2} a^2$$

The fourth term = $$^{n}C_3 x^{n – 3} a^3$$,  and so on.

We thus observe that the suffix of C in any term is one less than the number of terms, the index of x is n minus the suffix of C and the index of a is the same as the suffix of C.

Hence, the (r + 1)th term is given by $$^{n}C_r x^{n – r} a^r$$

Thus, if $$T_{r + 1}$$ denotes the (r + 1)th term, then

#### General Term :

$$T_{r + 1}$$ = $$^{n}C_r x^{n – r} a^r$$

This is called the general term, because by giving different values to r we can determine all terms of the expansion.

In the binomial expansion of $$(x – a)^n$$, the general term is given by

$$T_{r + 1}$$ = $$(-1)^r$$$$^{n}C_r x^{n – r} a^r$$

In the binomial expansion of $$(1 + x)^n$$, we have

$$T_{r + 1}$$ = $$^nC_r x^r$$

In the binomial expansion of $$(1 – x)^n$$, we have

$$T_{r + 1}$$ = $$(-1)^r$$$$^nC_r x^r$$

#### Nth term from the End :

In the binomial expansion of $$(x + a)^n$$, the rth term from the end is ((n + 1) – r + 1) = (n – r + 2)th term form the beginning.

Example : Write the general term in the expansion of $$(x^2 – y)^6$$.

Solution : We have, $$(x^2 – y)^6$$ = $$|(x^2 + (-y)|^6$$

The general term in the expansion of the above binomial is given by

$$T_{r + 1}$$ = $$^{n}C_r x^{n – r} a^r$$

$$\implies$$ $$T_{r + 1}$$ = $$^{6}C_r (x^2)^{6 – r} (-y)^r$$

$$\implies$$ $$T_{r + 1}$$ = $$(-1)^r$$$$^{6}C_r x^{12 – 2r} y^r$$

### Related Questions

Find the 9th term in the expansion of $$({x\over a} – {3a\over x^2})^{12}$$.

Find the 10th term in the binomial expansion of $$(2x^2 + {1\over x})^{12}$$.