7.2 Q-19 | Mathematical Induction And Binomial Theorem | Mathematics 11

Exercise 7.2 — Question 19

Find the middle term(s) in the expansion of $(x + \frac{1}{x})^{10}$ .

Step 1: Identify $n$ .

Here $n = 10$ , which is even.

Step 2: Determine the middle term.

Since $n$ is even, there is exactly one middle term: $T_{\frac{n}{2} + 1} = T_{\frac{10}{2} + 1} = T_{6}$

Step 3: Apply the general term formula.

The general term of $(a + b)^{n}$ is: $T_{r + 1} = (r n) a^{n - r} b^{r}$

For $T_{6}$ , set $r = 5$ , $a = x$ , $b = \frac{1}{x}$ , $n = 10$ : $T_{6} = (5 10) x^{10 - 5} (\frac{1}{x})^{5}$

Step 4: Simplify. $T_{6} = (5 10) x^{5} \cdot \frac{1}{x ^{5}} = (5 10) \cdot x^{0} = 252$

$T_{6} = 252$