Calculus Examples

$\sum_{3}^{14} 128 (\frac{1}{4})$

Step 1

Simplify the summation.

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Step 1.1

Cancel the common factor of $4$ .

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Step 1.1.1

Factor $4$ out of $128$ .

$4 (32) \frac{1}{4}$

Step 1.1.2

Cancel the common factor.

$4 \cdot 32 \frac{1}{4}$

Step 1.1.3

Rewrite the expression.

$32$

Step 1.2

Rewrite the summation.

$\sum_{i = 3}^{14} 32$

Step 2

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

$\sum_{i = 3}^{14} 32 = \sum_{i = 1}^{14} 32 - \sum_{i = 1}^{2} 32$

Step 3

Evaluate $\sum_{i = 1}^{14} 32$ .

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Step 3.1

The formula for the summation of a constant is:

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

Step 3.2

Substitute the values into the formula.

$(32) (14)$

Step 3.3

Multiply $32$ by $14$ .

$448$

Step 4

Evaluate $\sum_{i = 1}^{2} 32$ .

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Step 4.1

Expand the series for each value of $i$ .

$32 + 32$

Step 4.2

Add $32$ and $32$ .

$64$

Step 5

Replace the summations with the values found.

$448 - 64$

Step 6

Subtract $64$ from $448$ .

$384$