Question

Sigma Notation Write the sum using sigma notation.

$1-2x+3x^2-4x^3+5x^4+\mathrm{....}-100x^{99}.$

Short answer

The sum using sigma notation is$\sum _{n=1}^{100}{\left(-1\right)}^{n+1}n{x}^{n-1}.$

Step-by-step solution

Step 1.  Given

The given sequence is$1-2x+3x^2-4x^3+5x^4+\mathrm{....}-100x^{99}.$

Step 2.  To determine

We have to write the sum using sigma notation.

Step 3.  Calculation

We try to write the given sequence using a general term with n.

First term is = $1=1x^0={\left(-1\right)}^2{x}^0={\left(-1\right)}^{1+1}1x^{1-1}$

Second term is = $-2x=-2x^1=\left(-1\right)2x^1={\left(-1\right)}^{3}2x^1={\left(-1\right)}^{2+1}2x^{2-1}$

Third term is = $3x^2={\left(-1\right)}^{4}3x^2={\left(-1\right)}^{3+1}3x^{3-1}$

Fourth term is = $-4x^3=\left(-1\right)4x^3={\left(-1\right)}^{5}4x^3={\left(-1\right)}^{4+1}4x^{4-1}$

So, we see a pattern that the nth term is ${\left(-1\right)}^{n+1}n{x}^{n-1}$.

And n goes from 1 to 100.

Hence, the sum is given by: $\sum _{n=1}^{100}{\left(-1\right)}^{n+1}n{x}^{n-1}$.