Calculus Examples

$\sum_{i = 1}^{25} {(i^{2} + 2)}^{2}$

Step 1

Simplify the summation.

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Step 1.1

Rewrite ${(i^{2} + 2)}^{2}$ as $(i^{2} + 2) (i^{2} + 2)$ .

$(i^{2} + 2) (i^{2} + 2)$

Step 1.2

Expand $(i^{2} + 2) (i^{2} + 2)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$i^{2} (i^{2} + 2) + 2 (i^{2} + 2)$

Step 1.2.2

Apply the distributive property.

$i^{2} i^{2} + i^{2} \cdot 2 + 2 (i^{2} + 2)$

Step 1.2.3

Apply the distributive property.

$i^{2} i^{2} + i^{2} \cdot 2 + 2 i^{2} + 2 \cdot 2$

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Multiply $i^{2}$ by $i^{2}$ by adding the exponents.

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Step 1.3.1.1.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$i^{2 + 2} + i^{2} \cdot 2 + 2 i^{2} + 2 \cdot 2$

Step 1.3.1.1.2

Add $2$ and $2$ .

$i^{4} + i^{2} \cdot 2 + 2 i^{2} + 2 \cdot 2$

Step 1.3.1.2

Move $2$ to the left of $i^{2}$ .

$i^{4} + 2 \cdot i^{2} + 2 i^{2} + 2 \cdot 2$

Step 1.3.1.3

Multiply $2$ by $2$ .

$i^{4} + 2 i^{2} + 2 i^{2} + 4$

Step 1.3.2

Add $2 i^{2}$ and $2 i^{2}$ .

$i^{4} + 4 i^{2} + 4$

Step 1.4

Rewrite the summation.

$\sum_{i = 1}^{25} i^{4} + 4 i^{2} + 4$

Step 2

Split the summation into smaller summations that fit the summation rules.

$\sum_{i = 1}^{25} i^{4} + 4 i^{2} + 4 = \sum_{i = 1}^{25} i^{4} + 4 \sum_{i = 1}^{25} i^{2} + \sum_{i = 1}^{25} 4$

Step 3

Evaluate $\sum_{i = 1}^{25} i^{4}$ .

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Step 3.1

Expand the series for each value of $i$ .

$1^{4} + 2^{4} + 3^{4} + \dots + 25^{4}$

Step 3.2

Simplify the expanded form.

$2153645$

Step 4

Evaluate $4 \sum_{i = 1}^{25} i^{2}$ .

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Step 4.1

The formula for the summation of a polynomial with degree $2$ is:

$\sum_{k = 1}^{n} i^{2} = \frac{n (n + 1) (2 n + 1)}{6}$

Step 4.2

Substitute the values into the formula and make sure to multiply by the front term.

$(4) (\frac{25 (25 + 1) (2 \cdot 25 + 1)}{6})$

Step 4.3

Simplify.

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Step 4.3.1

Simplify the numerator.

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Step 4.3.1.1

Add $25$ and $1$ .

$4 \frac{25 \cdot 26 (2 \cdot 25 + 1)}{6}$

Step 4.3.1.2

Combine exponents.

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Step 4.3.1.2.1

Multiply $25$ by $26$ .

$4 \frac{650 (2 \cdot 25 + 1)}{6}$

Step 4.3.1.2.2

Multiply $2$ by $25$ .

$4 \frac{650 (50 + 1)}{6}$

Step 4.3.1.3

Add $50$ and $1$ .

$4 \frac{650 \cdot 51}{6}$

Step 4.3.2

Simplify terms.

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Step 4.3.2.1

Multiply $650$ by $51$ .

$4 (\frac{33150}{6})$

Step 4.3.2.2

Cancel the common factor of $2$ .

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Step 4.3.2.2.1

Factor $2$ out of $4$ .

$2 (2) \frac{33150}{6}$

Step 4.3.2.2.2

Factor $2$ out of $6$ .

$2 \cdot 2 \frac{33150}{2 \cdot 3}$

Step 4.3.2.2.3

Cancel the common factor.

$2 \cdot 2 \frac{33150}{2 \cdot 3}$

Step 4.3.2.2.4

Rewrite the expression.

$2 (\frac{33150}{3})$

Step 4.3.2.3

Combine $2$ and $\frac{33150}{3}$ .

$\frac{2 \cdot 33150}{3}$

Step 4.3.2.4

Simplify the expression.

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Step 4.3.2.4.1

Multiply $2$ by $33150$ .

$\frac{66300}{3}$

Step 4.3.2.4.2

Divide $66300$ by $3$ .

$22100$

Step 5

Evaluate $\sum_{i = 1}^{25} 4$ .

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Step 5.1

The formula for the summation of a constant is:

$\sum_{k = 1}^{n} c = c n$

Step 5.2

Substitute the values into the formula.

$(4) (25)$

Step 5.3

Multiply $4$ by $25$ .

$100$

Step 6

Add the results of the summations.

$2153645 + 22100 + 100$

Step 7

Simplify.

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Step 7.1

Add $2153645$ and $22100$ .

$2175745 + 100$

Step 7.2

Add $2175745$ and $100$ .

$2175845$