7.2 Q-3 | Mathematical Induction And Binomial Theorem | Mathematics 11

Exercise 7.2 — Question 3

This question applies the Binomial Theorem to find specific terms (including middle terms) in a binomial expansion.

For the expansion of $(a + b)^{n}$ , the general term (the $(r + 1)$ -th term) is:

$T_{r + 1} = (r n) a^{n - r} b^{r}, r = 0, 1, 2, \dots, n$

The total number of terms in $(a + b)^{n}$ is $n + 1$ .

If $n$ is even: there is one middle term, $T_{\frac{n}{2} + 1}$ .
If $n$ is odd: there are two middle terms, $T_{\frac{n + 1}{2}}$ and $T_{\frac{n + 3}{2}}$ .

Find the middle term(s) in the expansion of $(x + \frac{1}{x})^{6}$ .

Here $n = 6$ (even), so there is one middle term: $T_{\frac{6}{2} + 1} = T_{4}$ .

$T_{4} = T_{3 + 1} = (3 6) x^{6 - 3} (\frac{1}{x})^{3} = 20 \cdot x^{3} \cdot \frac{1}{x ^{3}} = 20$

Find the middle term(s) in the expansion of $(a + b)^{7}$ .

Here $n = 7$ (odd), so there are two middle terms: $T_{4}$ and $T_{5}$ .

$T_{4} = (3 7) a^{4} b^{3} = 35 a^{4} b^{3}$

$T_{5} = (4 7) a^{3} b^{4} = 35 a^{3} b^{4}$