Calculus Examples

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

Step 1

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

$\sum_{k = 2}^{24} 5 i^{2} + 8 = \sum_{k = 1}^{24} 5 i^{2} + 8 - \sum_{k = 1}^{1} 5 i^{2} + 8$

Step 2

Evaluate $\sum_{k = 1}^{24} 5 i^{2} + 8$ .

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Step 2.1

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

$\sum_{k = 1}^{24} 5 i^{2} + 8 = 5 \sum_{k = 1}^{24} i^{2} + \sum_{k = 1}^{24} 8$

Step 2.2

Evaluate $5 \sum_{k = 1}^{24} i^{2}$ .

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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 and make sure to multiply by the front term.

$(5) (\frac{24 (24 + 1) (2 \cdot 24 + 1)}{6})$

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 $24$ and $1$ .

$5 \frac{24 \cdot 25 (2 \cdot 24 + 1)}{6}$

Step 2.2.3.1.2

Combine exponents.

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Step 2.2.3.1.2.1

Multiply $24$ by $25$ .

$5 \frac{600 (2 \cdot 24 + 1)}{6}$

Step 2.2.3.1.2.2

Multiply $2$ by $24$ .

$5 \frac{600 (48 + 1)}{6}$

Step 2.2.3.1.3

Add $48$ and $1$ .

$5 \frac{600 \cdot 49}{6}$

Step 2.2.3.2

Simplify the expression.

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Step 2.2.3.2.1

Multiply $600$ by $49$ .

$5 (\frac{29400}{6})$

Step 2.2.3.2.2

Divide $29400$ by $6$ .

$5 \cdot 4900$

Step 2.2.3.2.3

Multiply $5$ by $4900$ .

$24500$

Step 2.3

Evaluate $\sum_{k = 1}^{24} 8$ .

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Step 2.3.1

The formula for the summation of a constant is:

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

Step 2.3.2

Substitute the values into the formula.

$(8) (24)$

Step 2.3.3

Multiply $8$ by $24$ .

$192$

Step 2.4

Add the results of the summations.

$24500 + 192$

Step 2.5

Add $24500$ and $192$ .

$24692$

Step 3

Evaluate $\sum_{k = 1}^{1} 5 i^{2} + 8$ .

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Step 3.1

Expand the series for each value of $i$ .

$5 \cdot 1^{2} + 8$

Step 3.2

Simplify.

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Step 3.2.1

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

$5 \cdot 1 + 8$

Step 3.2.2

Multiply $5$ by $1$ .

$5 + 8$

Step 3.2.3

Add $5$ and $8$ .

$13$

Step 4

Replace the summations with the values found.

$24692 - 13$

Step 5

Subtract $13$ from $24692$ .

$24679$