Summary

This chapter developed the binomial theorem for a positive integral index from first principles. Starting from a brief history of the coefficient pattern -- known in ancient India as Meru-prastara and in China through Jia Xian and Yang Hui, before Pascal's 1654 treatise gave it its common Western name -- the chapter stated the theorem

and gave a complete proof by mathematical induction, with the inductive step resting on the identity $^{n}C_{r-1}+{}^{n}C_r={}^{n+1}C_r$. That same identity is the construction rule of Pascal's Triangle, the triangular array whose row $n$ lists the coefficients $^{n}C_0,\ldots,{}^{n}C_n$ directly. The general term $T_{r+1}={}^{n}C_r\,a^{n-r}b^r$ was introduced to find any single term (or the coefficient of any specified power) without expanding the whole binomial, and was then specialised to locate the middle term -- a single term $T_{n/2+1}$ when $n$ is even, or two terms $T_{(n+1)/2}$ and $T_{(n+3)/2}$ when $n$ is odd. Finally, the chapter covered simple applications: truncating $(1+x)^n$ to approximate a power close to $1$, usi …