7.3 Q-4 | Mathematical Induction And Binomial Theorem | Mathematics 11

Exercise 7.3 — Question 4

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 $(r n) = \frac{n !}{r ! ( n - r )!}$ is the binomial coefficient.

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

$T_{r + 1} = (r n) a^{n - r} b^{r}$

Expand $(x + 2)^{4}$ using the Binomial Theorem.

Using $(a + b)^{n}$ with $a = x$ , $b = 2$ , $n = 4$ :

$(x + 2)^{4} = (0 4) x^{4} (2)^{0} + (1 4) x^{3} (2)^{1} + (2 4) x^{2} (2)^{2} + (3 4) x^{1} (2)^{3} + (4 4) x^{0} (2)^{4}$

$= 1 \cdot x^{4} + 4 \cdot 2 x^{3} + 6 \cdot 4 x^{2} + 4 \cdot 8 x + 1 \cdot 16$

$= x^{4} + 8 x^{3} + 24 x^{2} + 32 x + 16$

The Binomial Theorem applies when $n \in N$ (positive integers).
The expansion of $(a + b)^{n}$ has exactly $n + 1$ terms.
Binomial coefficients $(r n)$ are symmetric: $(r n) = (n - r n)$ .
The sum of all binomial coefficients equals $2^{n}$ : $r = 0 \sum n (r n) = 2^{n}$ .