7.3 Q-5 | Mathematical Induction And Binomial Theorem | Mathematics 11

Exercise 7.3 — Question 5

Expand the following using the Binomial Theorem and simplify:

$(x + \frac{1}{x})^{6}$

The Binomial Theorem states that for any positive integer $n$ :

$(a + b)^{n} = \sum_{r = 0}^{n} (r n) a^{n - r} b^{r}$

where the binomial coefficient is:

$(r n) = \frac{n !}{r ! ( n - r )!}$

Here $a = x$ , $b = \frac{1}{x}$ , and $n = 6$ .

Using the Binomial Theorem:

$(x + \frac{1}{x})^{6} = \sum_{r = 0}^{6} (r 6) x^{6 - r} (\frac{1}{x})^{r}$

Each term simplifies as:

$T_{r + 1} = (r 6) x^{6 - r} \cdot x^{- r} = (r 6) x^{6 - 2 r}$

Now compute each term:

$r$	$(r 6)$	Power of $x$	Term
0	1	$x^{6}$	$x^{6}$
1	6	$x^{4}$	$6 x^{4}$
2	15	$x^{2}$	$15 x^{2}$
3	20	$x^{0}$	$20$
4	15	$x^{- 2}$	$\frac{15}{x ^{2}}$
5	6	$x^{- 4}$	$\frac{6}{x ^{4}}$
6	1	$x^{- 6}$	$\frac{1}{x ^{6}}$

Final Answer:

$(x + \frac{1}{x})^{6} = x^{6} + 6 x^{4} + 15 x^{2} + 20 + \frac{15}{x ^{2}} + \frac{6}{x ^{4}} + \frac{1}{x ^{6}}$

The binomial coefficients $1, 6, 15, 20, 15, 6, 1$ can also be read directly from Row 6 of Pascal's Triangle:

$111615141513101262013101415151611$