Question

In the binomial expansion of \((1+x)^n\), the coefficients of \(x^{k-1}\), \(x^k\), \(x^{k+1}\) and also the coefficients of \(x^{l-1}\), \(x^l\), \(x^{l+1}\) are in A.P.

Then, the numerically greatest term in the expansion of \((1+x)^{14}\) when \(x=\dfrac{2}{3}\) is

• A. \({}^{14}C_{5}\left(\dfrac23\right)^5\)
• B. \({}^{14}C_{4}\left(\dfrac23\right)^4\)
• C. \({}^{15}C_{7}\left(\dfrac23\right)^8\)
• D. \({}^{15}C_{6}\left(\dfrac23\right)^9\)

Hint:

For the expansion of \((1+x)^n\), the greatest term is obtained by comparing successive terms using \[ \frac{T_{r+2}}{T_{r+1}} =\frac{n-r}{r+1}\,x. \] The last term for which this ratio is at least \(1\) is the numerically greatest term.

Answer 1

The Correct Option is A