Precalculus Examples

$\sum_{i = 24}^{83} 6 + 3 i$

Step 1

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

$\sum_{k = 24}^{83} 6 + 3 i = \sum_{k = 1}^{83} 6 + 3 i - \sum_{k = 1}^{23} 6 + 3 i$

Step 2

Evaluate $\sum_{k = 1}^{83} 6 + 3 i$ .

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Step 2.1

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

$\sum_{k = 1}^{83} 6 + 3 i = \sum_{k = 1}^{83} 6 + 3 \sum_{k = 1}^{83} i$

Step 2.2

Evaluate $\sum_{k = 1}^{83} 6$ .

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Step 2.2.1

The formula for the summation of a constant is:

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

Step 2.2.2

Substitute the values into the formula.

$(6) (83)$

Step 2.2.3

Multiply $6$ by $83$ .

$498$

Step 2.3

Evaluate $3 \sum_{k = 1}^{83} i$ .

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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.

$(3) (\frac{83 (83 + 1)}{2})$

Step 2.3.3

Simplify.

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Step 2.3.3.1

Add $83$ and $1$ .

$3 \frac{83 \cdot 84}{2}$

Step 2.3.3.2

Multiply $83$ by $84$ .

$3 (\frac{6972}{2})$

Step 2.3.3.3

Divide $6972$ by $2$ .

$3 \cdot 3486$

Step 2.3.3.4

Multiply $3$ by $3486$ .

$10458$

Step 2.4

Add the results of the summations.

$498 + 10458$

Step 2.5

Add $498$ and $10458$ .

$10956$

Step 3

Evaluate $\sum_{k = 1}^{23} 6 + 3 i$ .

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Step 3.1

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

$\sum_{k = 1}^{23} 6 + 3 i = \sum_{k = 1}^{23} 6 + 3 \sum_{k = 1}^{23} i$

Step 3.2

Evaluate $\sum_{k = 1}^{23} 6$ .

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Step 3.2.1

The formula for the summation of a constant is:

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

Step 3.2.2

Substitute the values into the formula.

$(6) (23)$

Step 3.2.3

Multiply $6$ by $23$ .

$138$

Step 3.3

Evaluate $3 \sum_{k = 1}^{23} i$ .

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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.

$(3) (\frac{23 (23 + 1)}{2})$

Step 3.3.3

Simplify.

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Step 3.3.3.1

Add $23$ and $1$ .

$3 \frac{23 \cdot 24}{2}$

Step 3.3.3.2

Multiply $23$ by $24$ .

$3 (\frac{552}{2})$

Step 3.3.3.3

Divide $552$ by $2$ .

$3 \cdot 276$

Step 3.3.3.4

Multiply $3$ by $276$ .

$828$

Step 3.4

Add the results of the summations.

$138 + 828$

Step 3.5

Add $138$ and $828$ .

$966$

Step 4

Replace the summations with the values found.

$10956 - 966$

Step 5

Subtract $966$ from $10956$ .

$9990$