Question

Sigma Notation Write the sum using sigma notation.

$1+x+x^2+x^3+\mathrm{.....}+x^{100}$.

Short answer

The sum using sigma notation is$\sum _{k=0}^{100}{x}^k.$

Step-by-step solution

Step 1.  Given

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

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 k.

$\begin{array}{l}1+x+x^2+x^3+\mathrm{.....}+x^{100}\\ =x^0+x^1+x^2+x^3+\mathrm{.....}+x^{100}\end{array}$

From the sequence it is easy to see that the kth term is $x^k$.

And k goes from 0 to 100.

Hence, the sum is given by: $\sum _{k=0}^{100}{x}^k$.