Calculus Examples

$\sum_{n = 1}^{7} 2 + n^{2}$

Step 1

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

$\sum_{n = 1}^{7} 2 + n^{2} = \sum_{n = 1}^{7} 2 + \sum_{n = 1}^{7} n^{2}$

Step 2

Evaluate $\sum_{n = 1}^{7} 2$ .

Tap for more steps...

Step 2.1

The formula for the summation of a constant is:

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

Step 2.2

Substitute the values into the formula.

$(2) (7)$

Step 2.3

Multiply $2$ by $7$ .

$14$

Step 3

Evaluate $\sum_{n = 1}^{7} n^{2}$ .

Tap for more steps...

Step 3.1

The formula for the summation of a polynomial with degree $2$ is:

$\sum_{k = 1}^{n} k^{2} = \frac{n (n + 1) (2 n + 1)}{6}$

Step 3.2

Substitute the values into the formula.

$\frac{7 (7 + 1) (2 \cdot 7 + 1)}{6}$

Step 3.3

Simplify.

Tap for more steps...

Step 3.3.1

Simplify the numerator.

Tap for more steps...

Step 3.3.1.1

Add $7$ and $1$ .

$\frac{7 \cdot 8 (2 \cdot 7 + 1)}{6}$

Step 3.3.1.2

Combine exponents.

Tap for more steps...

Step 3.3.1.2.1

Multiply $7$ by $8$ .

$\frac{56 (2 \cdot 7 + 1)}{6}$

Step 3.3.1.2.2

Multiply $2$ by $7$ .

$\frac{56 (14 + 1)}{6}$

Step 3.3.1.3

Add $14$ and $1$ .

$\frac{56 \cdot 15}{6}$

Step 3.3.2

Simplify the expression.

Tap for more steps...

Step 3.3.2.1

Multiply $56$ by $15$ .

$\frac{840}{6}$

Step 3.3.2.2

Divide $840$ by $6$ .

$140$

Step 4

Add the results of the summations.

$14 + 140$

Step 5

Add $14$ and $140$ .

$154$