Calculus Examples

$\sum_{i = 2}^{7} i^{2} - 2 i$

Step 1

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

$\sum_{k = 2}^{7} i^{2} - 2 i = \sum_{k = 1}^{7} i^{2} - 2 i - \sum_{k = 1}^{1} i^{2} - 2 i$

Step 2

Evaluate $\sum_{k = 1}^{7} i^{2} - 2 i$ .

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Step 2.1

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

$\sum_{k = 1}^{7} i^{2} - 2 i = \sum_{k = 1}^{7} i^{2} - 2 \sum_{k = 1}^{7} i$

Step 2.2

Evaluate $\sum_{k = 1}^{7} i^{2}$ .

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Step 2.2.1

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 2.2.2

Substitute the values into the formula.

$\frac{7 (7 + 1) (2 \cdot 7 + 1)}{6}$

Step 2.2.3

Simplify.

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Step 2.2.3.1

Simplify the numerator.

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Step 2.2.3.1.1

Add $7$ and $1$ .

$\frac{7 \cdot 8 (2 \cdot 7 + 1)}{6}$

Step 2.2.3.1.2

Combine exponents.

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Step 2.2.3.1.2.1

Multiply $7$ by $8$ .

$\frac{56 (2 \cdot 7 + 1)}{6}$

Step 2.2.3.1.2.2

Multiply $2$ by $7$ .

$\frac{56 (14 + 1)}{6}$

Step 2.2.3.1.3

Add $14$ and $1$ .

$\frac{56 \cdot 15}{6}$

Step 2.2.3.2

Simplify the expression.

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Step 2.2.3.2.1

Multiply $56$ by $15$ .

$\frac{840}{6}$

Step 2.2.3.2.2

Divide $840$ by $6$ .

$140$

Step 2.3

Evaluate $2 \sum_{k = 1}^{7} i$ .

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Step 2.3.1

The formula for the summation of a polynomial with degree $1$ is:

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

Step 2.3.2

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

$(- 2) (\frac{7 (7 + 1)}{2})$

Step 2.3.3

Simplify.

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Step 2.3.3.1

Simplify the expression.

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Step 2.3.3.1.1

Add $7$ and $1$ .

$- 2 \frac{7 \cdot 8}{2}$

Step 2.3.3.1.2

Multiply $7$ by $8$ .

$- 2 (\frac{56}{2})$

Step 2.3.3.2

Cancel the common factor of $2$ .

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Step 2.3.3.2.1

Factor $2$ out of $- 2$ .

$2 (- 1) \frac{56}{2}$

Step 2.3.3.2.2

Cancel the common factor.

$2 \cdot - 1 \frac{56}{2}$

Step 2.3.3.2.3

Rewrite the expression.

$- 1 \cdot 56$

Step 2.3.3.3

Multiply $- 1$ by $56$ .

$- 56$

Step 2.4

Add the results of the summations.

$140 - 56$

Step 2.5

Subtract $56$ from $140$ .

$84$

Step 3

Evaluate $\sum_{k = 1}^{1} i^{2} - 2 i$ .

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Step 3.1

Expand the series for each value of $i$ .

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

Step 3.2

Simplify.

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Step 3.2.1

Raise $1$ to the power of $2$ .

$1 - 2 \cdot 1$

Step 3.2.2

Multiply $- 2$ by $1$ .

$1 - 2$

Step 3.2.3

Subtract $2$ from $1$ .

$- 1$

Step 4

Replace the summations with the values found.

$84 + 1$

Step 5

Add $84$ and $1$ .

$85$