Numerically greatest term in the expansion of (x + a) n

Let T r and T r+1 be r th and (r +1) th terms respectively in the expansion of binomial (x+a) n . Then T r = n C r–1 x n–r+1 a r–1 and T r+1 = n C r x n–r a r ∴ =

Now, T r+1 > , = , < T r According as > , =

, < 1, i.e., according as –1 >, = , < , i.e. according

as r < , = , >

So, if is an integer, say p, then T r+1 > T r

if r < p otherwise T r+1 ≤ T r

So, T p = T p+1 (numerically) and these are greater than any other term in the expansion.

Next, if is a non- integer, suppose m be its

integral part then T r+1 < T r if r ≤ m and T r+1 < T r if r > m.

So, T m+1 is the numerically greatest term among the terms of the expansion.

Again we can also write that k

th term is numerically greatest if T k > T k+1 and T k > T k–1 .

(i) The numerically greatest term in the expansion of

(1–2x) 8 , when x = 2 is –