Calculus Examples

$\sum_{i = 5}^{24} 2 i^{2} - i + 3 + 8$

Step 1

Simplify the summation.

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Step 1.1

Add $3$ and $8$ .

$2 i^{2} - i + 11$

Step 1.2

Rewrite the summation.

$\sum_{i = 5}^{24} 2 i^{2} - i + 11$

Step 2

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

$\sum_{i = 5}^{24} 2 i^{2} - i + 11 = \sum_{i = 1}^{24} 2 i^{2} - i + 11 - \sum_{i = 1}^{4} 2 i^{2} - i + 11$

Step 3

Evaluate $\sum_{i = 1}^{24} 2 i^{2} - i + 11$ .

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Step 3.1

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

$\sum_{k = 1}^{24} 2 i^{2} - i + 11 = 2 \sum_{k = 1}^{24} i^{2} - 1 \sum_{k = 1}^{24} i + \sum_{k = 1}^{24} 11$

Step 3.2

Evaluate $2 \sum_{k = 1}^{24} i^{2}$ .

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Step 3.2.1

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 3.2.2

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

$(2) (\frac{24 (24 + 1) (2 \cdot 24 + 1)}{6})$

Step 3.2.3

Simplify.

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Step 3.2.3.1

Simplify the numerator.

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Step 3.2.3.1.1

Add $24$ and $1$ .

$2 \frac{24 \cdot 25 (2 \cdot 24 + 1)}{6}$

Step 3.2.3.1.2

Combine exponents.

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Step 3.2.3.1.2.1

Multiply $24$ by $25$ .

$2 \frac{600 (2 \cdot 24 + 1)}{6}$

Step 3.2.3.1.2.2

Multiply $2$ by $24$ .

$2 \frac{600 (48 + 1)}{6}$

Step 3.2.3.1.3

Add $48$ and $1$ .

$2 \frac{600 \cdot 49}{6}$

Step 3.2.3.2

Reduce the expression by cancelling the common factors.

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Step 3.2.3.2.1

Multiply $600$ by $49$ .

$2 (\frac{29400}{6})$

Step 3.2.3.2.2

Cancel the common factor of $2$ .

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Step 3.2.3.2.2.1

Factor $2$ out of $6$ .

$2 \frac{29400}{2 (3)}$

Step 3.2.3.2.2.2

Cancel the common factor.

$2 \frac{29400}{2 \cdot 3}$

Step 3.2.3.2.2.3

Rewrite the expression.

$\frac{29400}{3}$

Step 3.2.3.2.3

Divide $29400$ by $3$ .

$9800$

Step 3.3

Evaluate $1 \sum_{k = 1}^{24} i$ .

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Step 3.3.1

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

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

Step 3.3.2

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

$(- 1) (\frac{24 (24 + 1)}{2})$

Step 3.3.3

Simplify.

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Step 3.3.3.1

Add $24$ and $1$ .

$- 1 \frac{24 \cdot 25}{2}$

Step 3.3.3.2

Multiply $24$ by $25$ .

$- 1 (\frac{600}{2})$

Step 3.3.3.3

Divide $600$ by $2$ .

$- 1 \cdot 300$

Step 3.3.3.4

Multiply $- 1$ by $300$ .

$- 300$

Step 3.4

Evaluate $\sum_{k = 1}^{24} 11$ .

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Step 3.4.1

The formula for the summation of a constant is:

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

Step 3.4.2

Substitute the values into the formula.

$(11) (24)$

Step 3.4.3

Multiply $11$ by $24$ .

$264$

Step 3.5

Add the results of the summations.

$9800 - 300 + 264$

Step 3.6

Simplify.

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Step 3.6.1

Subtract $300$ from $9800$ .

$9500 + 264$

Step 3.6.2

Add $9500$ and $264$ .

$9764$

Step 4

Evaluate $\sum_{i = 1}^{4} 2 i^{2} - i + 11$ .

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Step 4.1

Expand the series for each value of $i$ .

$2 \cdot 1^{2} - 1 \cdot 1 + 11 + 2 \cdot 2^{2} - 1 \cdot 2 + 11 + 2 \cdot 3^{2} - 1 \cdot 3 + 11 + 2 \cdot 4^{2} - 1 \cdot 4 + 11$

Step 4.2

Simplify.

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Step 4.2.1

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

$2 \cdot 1 - 1 \cdot 1 + 11 + 2 \cdot 2^{2} - 1 \cdot 2 + 11 + 2 \cdot 3^{2} - 1 \cdot 3 + 11 + 2 \cdot 4^{2} - 1 \cdot 4 + 11$

Step 4.2.2

Multiply $2$ by $1$ .

$2 - 1 \cdot 1 + 11 + 2 \cdot 2^{2} - 1 \cdot 2 + 11 + 2 \cdot 3^{2} - 1 \cdot 3 + 11 + 2 \cdot 4^{2} - 1 \cdot 4 + 11$

Step 4.2.3

Multiply $- 1$ by $1$ .

$2 - 1 + 11 + 2 \cdot 2^{2} - 1 \cdot 2 + 11 + 2 \cdot 3^{2} - 1 \cdot 3 + 11 + 2 \cdot 4^{2} - 1 \cdot 4 + 11$

Step 4.2.4

Subtract $1$ from $2$ .

$1 + 11 + 2 \cdot 2^{2} - 1 \cdot 2 + 11 + 2 \cdot 3^{2} - 1 \cdot 3 + 11 + 2 \cdot 4^{2} - 1 \cdot 4 + 11$

Step 4.2.5

Add $1$ and $11$ .

$12 + 2 \cdot 2^{2} - 1 \cdot 2 + 11 + 2 \cdot 3^{2} - 1 \cdot 3 + 11 + 2 \cdot 4^{2} - 1 \cdot 4 + 11$

Step 4.2.6

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

$12 + 2 \cdot 4 - 1 \cdot 2 + 11 + 2 \cdot 3^{2} - 1 \cdot 3 + 11 + 2 \cdot 4^{2} - 1 \cdot 4 + 11$

Step 4.2.7

Multiply $2$ by $4$ .

$12 + 8 - 1 \cdot 2 + 11 + 2 \cdot 3^{2} - 1 \cdot 3 + 11 + 2 \cdot 4^{2} - 1 \cdot 4 + 11$

Step 4.2.8

Multiply $- 1$ by $2$ .

$12 + 8 - 2 + 11 + 2 \cdot 3^{2} - 1 \cdot 3 + 11 + 2 \cdot 4^{2} - 1 \cdot 4 + 11$

Step 4.2.9

Subtract $2$ from $8$ .

$12 + 6 + 11 + 2 \cdot 3^{2} - 1 \cdot 3 + 11 + 2 \cdot 4^{2} - 1 \cdot 4 + 11$

Step 4.2.10

Add $6$ and $11$ .

$12 + 17 + 2 \cdot 3^{2} - 1 \cdot 3 + 11 + 2 \cdot 4^{2} - 1 \cdot 4 + 11$

Step 4.2.11

Add $12$ and $17$ .

$29 + 2 \cdot 3^{2} - 1 \cdot 3 + 11 + 2 \cdot 4^{2} - 1 \cdot 4 + 11$

Step 4.2.12

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

$29 + 2 \cdot 9 - 1 \cdot 3 + 11 + 2 \cdot 4^{2} - 1 \cdot 4 + 11$

Step 4.2.13

Multiply $2$ by $9$ .

$29 + 18 - 1 \cdot 3 + 11 + 2 \cdot 4^{2} - 1 \cdot 4 + 11$

Step 4.2.14

Multiply $- 1$ by $3$ .

$29 + 18 - 3 + 11 + 2 \cdot 4^{2} - 1 \cdot 4 + 11$

Step 4.2.15

Subtract $3$ from $18$ .

$29 + 15 + 11 + 2 \cdot 4^{2} - 1 \cdot 4 + 11$

Step 4.2.16

Add $15$ and $11$ .

$29 + 26 + 2 \cdot 4^{2} - 1 \cdot 4 + 11$

Step 4.2.17

Add $29$ and $26$ .

$55 + 2 \cdot 4^{2} - 1 \cdot 4 + 11$

Step 4.2.18

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

$55 + 2 \cdot 16 - 1 \cdot 4 + 11$

Step 4.2.19

Multiply $2$ by $16$ .

$55 + 32 - 1 \cdot 4 + 11$

Step 4.2.20

Multiply $- 1$ by $4$ .

$55 + 32 - 4 + 11$

Step 4.2.21

Subtract $4$ from $32$ .

$55 + 28 + 11$

Step 4.2.22

Add $28$ and $11$ .

$55 + 39$

Step 4.2.23

Add $55$ and $39$ .

$94$

Step 5

Replace the summations with the values found.

$9764 - 94$

Step 6

Subtract $94$ from $9764$ .

$9670$