Calculus Examples

$\sum_{i = 1}^{5} {(2 i - 1)}^{2}$

Step 1

Simplify the summation.

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Step 1.1

Rewrite ${(2 i - 1)}^{2}$ as $(2 i - 1) (2 i - 1)$ .

$(2 i - 1) (2 i - 1)$

Step 1.2

Expand $(2 i - 1) (2 i - 1)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$2 i (2 i - 1) - 1 (2 i - 1)$

Step 1.2.2

Apply the distributive property.

$2 i (2 i) + 2 i \cdot - 1 - 1 (2 i - 1)$

Step 1.2.3

Apply the distributive property.

$2 i (2 i) + 2 i \cdot - 1 - 1 (2 i) - 1 \cdot - 1$

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

Rewrite using the commutative property of multiplication.

$2 \cdot 2 i \cdot i + 2 i \cdot - 1 - 1 (2 i) - 1 \cdot - 1$

Step 1.3.1.2

Multiply $i$ by $i$ by adding the exponents.

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Step 1.3.1.2.1

Move $i$ .

$2 \cdot 2 (i \cdot i) + 2 i \cdot - 1 - 1 (2 i) - 1 \cdot - 1$

Step 1.3.1.2.2

Multiply $i$ by $i$ .

$2 \cdot 2 i^{2} + 2 i \cdot - 1 - 1 (2 i) - 1 \cdot - 1$

Step 1.3.1.3

Multiply $2$ by $2$ .

$4 i^{2} + 2 i \cdot - 1 - 1 (2 i) - 1 \cdot - 1$

Step 1.3.1.4

Multiply $- 1$ by $2$ .

$4 i^{2} - 2 i - 1 (2 i) - 1 \cdot - 1$

Step 1.3.1.5

Multiply $2$ by $- 1$ .

$4 i^{2} - 2 i - 2 i - 1 \cdot - 1$

Step 1.3.1.6

Multiply $- 1$ by $- 1$ .

$4 i^{2} - 2 i - 2 i + 1$

Step 1.3.2

Subtract $2 i$ from $- 2 i$ .

$4 i^{2} - 4 i + 1$

Step 1.4

Rewrite the summation.

$\sum_{i = 1}^{5} 4 i^{2} - 4 i + 1$

Step 2

Expand the series for each value of $i$ .

$4 \cdot 1^{2} - 4 \cdot 1 + 1 + 4 \cdot 2^{2} - 4 \cdot 2 + 1 + 4 \cdot 3^{2} - 4 \cdot 3 + 1 + \dots + 4 \cdot 5^{2} - 4 \cdot 5 + 1$

Step 3

Simplify the expanded form.

$165$