Precalculus Examples

$\sum_{i = 1}^{4} \frac{1}{7} \cdot (i^{2} + 1)$

Step 1

Simplify the summation.

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Step 1.1

Apply the distributive property.

$\frac{1}{7} i^{2} + \frac{1}{7} \cdot 1$

Step 1.2

Combine $\frac{1}{7}$ and $i^{2}$ .

$\frac{i^{2}}{7} + \frac{1}{7} \cdot 1$

Step 1.3

Multiply $\frac{1}{7}$ by $1$ .

$\frac{i^{2}}{7} + \frac{1}{7}$

Step 1.4

Rewrite the summation.

$\sum_{i = 1}^{4} \frac{i^{2}}{7} + \frac{1}{7}$

Step 2

Expand the series for each value of $i$ .

$\frac{1^{2}}{7} + \frac{1}{7} + \frac{2^{2}}{7} + \frac{1}{7} + \frac{3^{2}}{7} + \frac{1}{7} + \frac{4^{2}}{7} + \frac{1}{7}$

Step 3

Simplify.

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Step 3.1

Combine the numerators over the common denominator.

$\frac{1^{2} + 1 + 2^{2} + 1 + 3^{2} + 1 + 4^{2} + 1}{7}$

Step 3.2

Simplify each term.

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Step 3.2.1

One to any power is one.

$\frac{1 + 1 + 2^{2} + 1 + 3^{2} + 1 + 4^{2} + 1}{7}$

Step 3.2.2

Raise $2$ to the power of $2$ .

$\frac{1 + 1 + 4 + 1 + 3^{2} + 1 + 4^{2} + 1}{7}$

Step 3.2.3

Raise $3$ to the power of $2$ .

$\frac{1 + 1 + 4 + 1 + 9 + 1 + 4^{2} + 1}{7}$

Step 3.2.4

Raise $4$ to the power of $2$ .

$\frac{1 + 1 + 4 + 1 + 9 + 1 + 16 + 1}{7}$

Step 3.3

Add $1$ and $1$ .

$\frac{2 + 4 + 1 + 9 + 1 + 16 + 1}{7}$

Step 3.4

Add $2$ and $4$ .

$\frac{6 + 1 + 9 + 1 + 16 + 1}{7}$

Step 3.5

Add $6$ and $1$ .

$\frac{7 + 9 + 1 + 16 + 1}{7}$

Step 3.6

Add $7$ and $9$ .

$\frac{16 + 1 + 16 + 1}{7}$

Step 3.7

Add $16$ and $1$ .

$\frac{17 + 16 + 1}{7}$

Step 3.8

Add $17$ and $16$ .

$\frac{33 + 1}{7}$

Step 3.9

Add $33$ and $1$ .

$\frac{34}{7}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{34}{7}$

Decimal Form:

$4.\overline{857142}$

Mixed Number Form:

$4 \frac{6}{7}$