Calculus Examples

$\sum_{i = 1}^{n} {(3 i - 1)}^{2}$

Step 1

Simplify the summation.

Tap for more steps...

Step 1.1

Rewrite ${(3 i - 1)}^{2}$ as $(3 i - 1) (3 i - 1)$ .

$(3 i - 1) (3 i - 1)$

Step 1.2

Expand $(3 i - 1) (3 i - 1)$ using the FOIL Method.

Tap for more steps...

Step 1.2.1

Apply the distributive property.

$3 i (3 i - 1) - 1 (3 i - 1)$

Step 1.2.2

Apply the distributive property.

$3 i (3 i) + 3 i \cdot - 1 - 1 (3 i - 1)$

Step 1.2.3

Apply the distributive property.

$3 i (3 i) + 3 i \cdot - 1 - 1 (3 i) - 1 \cdot - 1$

Step 1.3

Simplify and combine like terms.

Tap for more steps...

Step 1.3.1

Simplify each term.

Tap for more steps...

Step 1.3.1.1

Rewrite using the commutative property of multiplication.

$3 \cdot 3 i \cdot i + 3 i \cdot - 1 - 1 (3 i) - 1 \cdot - 1$

Step 1.3.1.2

Multiply $i$ by $i$ by adding the exponents.

Tap for more steps...

Step 1.3.1.2.1

Move $i$ .

$3 \cdot 3 (i \cdot i) + 3 i \cdot - 1 - 1 (3 i) - 1 \cdot - 1$

Step 1.3.1.2.2

Multiply $i$ by $i$ .

$3 \cdot 3 i^{2} + 3 i \cdot - 1 - 1 (3 i) - 1 \cdot - 1$

Step 1.3.1.3

Multiply $3$ by $3$ .

$9 i^{2} + 3 i \cdot - 1 - 1 (3 i) - 1 \cdot - 1$

Step 1.3.1.4

Multiply $- 1$ by $3$ .

$9 i^{2} - 3 i - 1 (3 i) - 1 \cdot - 1$

Step 1.3.1.5

Multiply $3$ by $- 1$ .

$9 i^{2} - 3 i - 3 i - 1 \cdot - 1$

Step 1.3.1.6

Multiply $- 1$ by $- 1$ .

$9 i^{2} - 3 i - 3 i + 1$

Step 1.3.2

Subtract $3 i$ from $- 3 i$ .

$9 i^{2} - 6 i + 1$

Step 1.4

Rewrite the summation.

$\sum_{i = 1}^{n} 9 i^{2} - 6 i + 1$

Step 2

The formula for the summation of a constant is:

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

Step 3

Substitute the values into the formula.

$(9 i^{2} - 6 i + 1) (n)$

Step 4

Simplify.

Tap for more steps...

Step 4.1

Simplify each term.

Tap for more steps...

Step 4.1.1

Rewrite $i^{2}$ as $- 1$ .

$(9 \cdot - 1 - 6 i + 1) n$

Step 4.1.2

Multiply $9$ by $- 1$ .

$(- 9 - 6 i + 1) n$

Step 4.2

Simplify by multiplying through.

Tap for more steps...

Step 4.2.1

Add $- 9$ and $1$ .

$(- 8 - 6 i) n$

Step 4.2.2

Apply the distributive property.

$- 8 n - 6 i n$