Binomial Theorem — Chapter Summary

For any positive integer \(n\)

For any positive integer \(n\), the expansion of \((x + y)^n\) is given by: \[(x + y)^n = {}^nC_0 x^n + {}^nC_1 x^{n-1}y + {}^nC_2 x^{n-2}y^2 + \cdots

The \((r+1)^{\text{th}}\) term in the expansion

The \((r+1)^{\text{th}}\) term in the expansion of \((x + y)^n\) is given by: \[T_{r+1} = {}^nC_r x^{n-r} y^r\] This formula helps in finding any spec

When \(n\) is even

When \(n\) is even, there is one middle term: the \(\left(\frac{n}{2} + 1\right)^{\text{th}}\) term. When \(n\) is odd, there are two middle terms: th

When the exponent \(r\) is

When the exponent \(r\) is a rational number and \(|x| < 1\), the expansion of \((1 + x)^r\) is an infinite series: \[(1 + x)^r = 1 + rx + \frac{r(r-1

For small values of \(x\)

For small values of \(x\), higher powers of \(x\) can be neglected. This allows us to approximate expressions like \((1 + x)^n\) using only the first