Question

8) Find and simplify the first 3 terms of (1 + x)^13.Use these first 3 terms to approximate (1.1)^13

Answer 1

ANSWER

$\begin{gathered} \text{The first thre}e\text{ terms of the binomial expression is given as} \\ 1+13x+78x^2 \end{gathered}$

STEP-BY-STEP EXPLANATION:

Given the following function

$(1+x)^{13}$

The above expression is a binomial expression

To find the first three terms, we need to apply the binomial theorem

The general formula for binomial theorem is given below as

$^nC_rx^{n\text{ - r}}y^r$

Let x = 1 and y = x

n = 13

r ranges from zero to 13

$\begin{gathered} ^{13}C_{0\cdot}1^{13\text{ - 0}}\cdot x^0+^{13}C^{}_1\cdot1^{13\text{ - 1}}\cdot x^1+^{13}C_2\cdot1^{13\text{ - 2}}x^2 \\ \text{ Recall that, the combination formula is given as} \\ ^nC_r\text{ = }\frac{n!}{(n\text{ - r)!r!}} \\ \frac{13!}{(13-0)!0!}\cdot1^{13\text{ - 0 }}\cdot x^0\text{ + }\frac{13!}{(13\text{ - 1)!1!}}\cdot1^{13\text{ -1}}\cdot x^1\text{ + }\frac{13!}{(13\text{ - 2)!2!}}\cdot1^{13\text{ - 2}}\cdot x^2 \\ \frac{13!}{13!}\cdot1^{13-\text{ 0}}\cdot x^0\text{ + }\frac{13!}{12!}\cdot1^{12}\cdot x^1\text{ + }\frac{13!}{11!2!}\cdot1^{13\text{ - 2}}\cdot x^2 \\ 1\cdot\text{ 1 }\cdot\text{ 1 + 13 }\cdot\text{ 1 }\cdot\text{ x + }78\cdot\text{ 1 }\cdot x^2 \\ 1+13x+78x^2 \end{gathered}$

Hence, the first three terms of the binomial expression are given below as

$1+13x+78x^2$