n^th term from the end of the expansion of\(\rm (2x - \frac {x} {2})^n\)

Question

Question

Answer 1

Concept:

General term: General term in the expansion of (x + y)n is given by

\(\rm {T_{\left( {r\; + \;1} \right)}} = {\;^n}{C_r} \times {x^{n - r}} \times {y^r}\)

In the expansion of (x + y)n the number of terms is (n + 1)

From the end (n + 1)^th term is 1st term and n^thterm is 2 term

Calculation:

In the expansion of \(\rm (2x - \frac {x} {2})^n\) ,n^th term from the end of an expansion is 2nd term

T_r+1 = ⁿC_r (2x)⁽^{n - r)} \(\rm (\frac {-x} {2})^r\)

T₂ = ⁿC₁.(2x)^{(n - 1)} \(\rm (\frac {-x} {2})\)

= \(\rm - n \times 2^{(n - 1 - 1)} \times x^{(n - 1 +1)}\)

= \(\rm - n \times 2^{(n - 2)} \times x^{n}\)

= \(\rm - nx^n. 2^{(n - 2)}\)