The binomial coefficients appearing in the expansion of are often said to "rise and…

MATHEMATICS · Subjective · A Level · Mathematics · Sequences and series · Sequences and series

Problem

The binomial coefficients appearing in the expansion of (a+b)n(a+b)^n are often said to "rise and then fall" as the index increases from 00 to nn. (i) Write the general expression for the kkth binomial coefficient in the expansion of (a+b)n(a+b)^n. (ii) Prove that for 0≤k<n0\le k<n the ratio of two consecutive coefficients is (nk+1)(nk)=n−kk+1.\frac{{n\choose k+1}}{{n\choose k}}=\frac{n-k}{k+1}. (iii) Using the result of part (ii), explain why the sequence of binomial coefficients first increases, reaches a maximum (or two equal maxima), and then decreases. Hence state precisely which coefficient(s) is/are numerically greatest for a given positive integer nn.

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