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

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

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!}} \\ (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 \\ (13!)/(13!)\cdot1^{13-\text{ 0}}\cdot x^0\text{ + }(13!)/(12!)\cdot1^(12)\cdot x^1\text{ + }(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`