Precalculus Examples

$\sum_{k = 4}^{110} x^{k}$

Step 1

Split the summation to make the starting value of $k$ equal to $1$ .

$\sum_{k = 4}^{110} x^{k} = \sum_{k = 1}^{110} x^{k} - \sum_{k = 1}^{3} x^{k}$

Step 2

Evaluate $\sum_{k = 1}^{110} x^{k}$ .

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Step 2.1

Expand the series for each value of $k$ .

$x^{1} + x^{2} + x^{3} + \dots + x^{110}$

Step 2.2

Simplify the expanded form.

$0$

Step 3

Expand the series for each value of $k$ .

$x^{1} + x^{2} + x^{3}$

Step 4

Replace the summations with the values found.

$0 - (x + x^{2} + x^{3})$

Step 5

Simplify.

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Step 5.1

Subtract $x^{1} + x^{2} + x^{3}$ from $0$ .

$- (x^{1} + x^{2} + x^{3})$

Step 5.2

Simplify.

$- (x + x^{2} + x^{3})$

Step 5.3

Apply the distributive property.

$- x - x^{2} - x^{3}$