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 \((2x^2 + {1\over x})^{12}\).
