Calculus Examples

$\sum_{n = 1}^{14} 3 n + 2$

Step 1

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

$\sum_{n = 1}^{14} 3 n + 2 = 3 \sum_{n = 1}^{14} n + \sum_{n = 1}^{14} 2$

Step 2

Evaluate $3 \sum_{n = 1}^{14} n$ .

Tap for more steps...

Step 2.1

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

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

Step 2.2

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

$(3) (\frac{14 (14 + 1)}{2})$

Step 2.3

Simplify.

Tap for more steps...

Step 2.3.1

Add $14$ and $1$ .

$3 \frac{14 \cdot 15}{2}$

Step 2.3.2

Multiply $14$ by $15$ .

$3 (\frac{210}{2})$

Step 2.3.3

Divide $210$ by $2$ .

$3 \cdot 105$

Step 2.3.4

Multiply $3$ by $105$ .

$315$

Step 3

Evaluate $\sum_{n = 1}^{14} 2$ .

Tap for more steps...

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.

$(2) (14)$

Step 3.3

Multiply $2$ by $14$ .

$28$

Step 4

Add the results of the summations.

$315 + 28$

Step 5

Add $315$ and $28$ .

$343$