<div class="content-box-placement content-box-placement--full content-box-placement--align-center" style="display:flex;justify-content:stretch;max-width:100%"><div class="content-box content-box--emphasis" data-content-box-preset="emphasis" style="box-sizing:border-box;border-color:var(--content-box-border, #1E6FBA);border-style:solid;border-width:0px;border-radius:0px;background-color:var(--content-box-background, #F5F8FC);color:var(--content-box-foreground, #111827);padding:12px 16px;font-family:'EB Garamond', serif;font-size:16px;line-height:1.35;width:100%;max-width:100%;border-left-width:3px"><p style="text-align: center">\((1+x)^n=1+nx+\dfrac{n(n-1)}2x^2+\cdots+x^n\qquad n\in\mathbb Z_{&gt;0}\)</p></div></div>
<p class="paragraph-header">Expansion in ascending powers</p>
<p style="text-align: center">\(\begin{aligned} (1+3x)^5&=1+\dbinom{5}{1}(3x)+\dbinom{5}{2}(3x)^2+\dbinom{5}{3}(3x)^3+\cdots \\ &=1+15x+90x^2+270x^3+\cdots \end{aligned}\)</p>
<p class="paragraph-header">Coefficient in a product</p>
<p>\((2-x)(1+3x)^5=(2-x)(1+15x+90x^2+\cdots)\)</p>
<p>Coefficient of \(x^2\): \(2(90)+(-1)(15)=165\)</p>
<p class="paragraph-header">Approximation</p>
<p>Substitute the given value into the expansion. Use enough terms for the required accuracy and round at the end.</p>
<p>With \(x=0.01\): \((1.03)^5\approx 1+15(0.01)+90(0.01)^2+270(0.01)^3=1.15927\), so \((1.03)^5\approx 1.159\).</p>

\((1+x)^n=1+nx+\dfrac{n(n-1)}2x^2+\cdots+x^n\qquad n\in\mathbb Z_{>0}\)

\(\begin{aligned} (1+3x)^5&=1+\dbinom{5}{1}(3x)+\dbinom{5}{2}(3x)^2+\dbinom{5}{3}(3x)^3+\cdots \\ &=1+15x+90x^2+270x^3+\cdots \end{aligned}\)

\((2-x)(1+3x)^5=(2-x)(1+15x+90x^2+\cdots)\)

Coefficient of \(x^2\): \(2(90)+(-1)(15)=165\)

Substitute the given value into the expansion. Use enough terms for the required accuracy and round at the end.

With \(x=0.01\): \((1.03)^5\approx 1+15(0.01)+90(0.01)^2+270(0.01)^3=1.15927\), so \((1.03)^5\approx 1.159\).

First Few Terms and Approximations

First Few Terms and Approximations