Calculus Examples

$\sum_{m = 1}^{4} {(3 - m)}^{2}$

Step 1

Simplify the summation.

Tap for more steps...

Step 1.1

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

$(3 - m) (3 - m)$

Step 1.2

Expand $(3 - m) (3 - m)$ using the FOIL Method.

Tap for more steps...

Step 1.2.1

Apply the distributive property.

$3 (3 - m) - m (3 - m)$

Step 1.2.2

Apply the distributive property.

$3 \cdot 3 + 3 (- m) - m (3 - m)$

Step 1.2.3

Apply the distributive property.

$3 \cdot 3 + 3 (- m) - m \cdot 3 - m (- m)$

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

Multiply $3$ by $3$ .

$9 + 3 (- m) - m \cdot 3 - m (- m)$

Step 1.3.1.2

Multiply $- 1$ by $3$ .

$9 - 3 m - m \cdot 3 - m (- m)$

Step 1.3.1.3

Multiply $3$ by $- 1$ .

$9 - 3 m - 3 m - m (- m)$

Step 1.3.1.4

Rewrite using the commutative property of multiplication.

$9 - 3 m - 3 m - 1 \cdot - 1 m \cdot m$

Step 1.3.1.5

Multiply $m$ by $m$ by adding the exponents.

Tap for more steps...

Step 1.3.1.5.1

Move $m$ .

$9 - 3 m - 3 m - 1 \cdot - 1 (m \cdot m)$

Step 1.3.1.5.2

Multiply $m$ by $m$ .

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

Step 1.3.1.6

Multiply $- 1$ by $- 1$ .

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

Step 1.3.1.7

Multiply $m^{2}$ by $1$ .

$9 - 3 m - 3 m + m^{2}$

Step 1.3.2

Subtract $3 m$ from $- 3 m$ .

$9 - 6 m + m^{2}$

Step 1.4

Rewrite the summation.

$\sum_{m = 1}^{4} 9 - 6 m + m^{2}$

Step 2

Expand the series for each value of $m$ .

$9 - 6 \cdot 1 + 1^{2} + 9 - 6 \cdot 2 + 2^{2} + 9 - 6 \cdot 3 + 3^{2} + 9 - 6 \cdot 4 + 4^{2}$

Step 3

Simplify.

Tap for more steps...

Step 3.1

Multiply $- 6$ by $1$ .

$9 - 6 + 1^{2} + 9 - 6 \cdot 2 + 2^{2} + 9 - 6 \cdot 3 + 3^{2} + 9 - 6 \cdot 4 + 4^{2}$

Step 3.2

Subtract $6$ from $9$ .

$3 + 1^{2} + 9 - 6 \cdot 2 + 2^{2} + 9 - 6 \cdot 3 + 3^{2} + 9 - 6 \cdot 4 + 4^{2}$

Step 3.3

Raise $1$ to the power of $2$ .

$3 + 1 + 9 - 6 \cdot 2 + 2^{2} + 9 - 6 \cdot 3 + 3^{2} + 9 - 6 \cdot 4 + 4^{2}$

Step 3.4

Add $3$ and $1$ .

$4 + 9 - 6 \cdot 2 + 2^{2} + 9 - 6 \cdot 3 + 3^{2} + 9 - 6 \cdot 4 + 4^{2}$

Step 3.5

Multiply $- 6$ by $2$ .

$4 + 9 - 12 + 2^{2} + 9 - 6 \cdot 3 + 3^{2} + 9 - 6 \cdot 4 + 4^{2}$

Step 3.6

Subtract $12$ from $9$ .

$4 - 3 + 2^{2} + 9 - 6 \cdot 3 + 3^{2} + 9 - 6 \cdot 4 + 4^{2}$

Step 3.7

Raise $2$ to the power of $2$ .

$4 - 3 + 4 + 9 - 6 \cdot 3 + 3^{2} + 9 - 6 \cdot 4 + 4^{2}$

Step 3.8

Add $- 3$ and $4$ .

$4 + 1 + 9 - 6 \cdot 3 + 3^{2} + 9 - 6 \cdot 4 + 4^{2}$

Step 3.9

Add $4$ and $1$ .

$5 + 9 - 6 \cdot 3 + 3^{2} + 9 - 6 \cdot 4 + 4^{2}$

Step 3.10

Multiply $- 6$ by $3$ .

$5 + 9 - 18 + 3^{2} + 9 - 6 \cdot 4 + 4^{2}$

Step 3.11

Subtract $18$ from $9$ .

$5 - 9 + 3^{2} + 9 - 6 \cdot 4 + 4^{2}$

Step 3.12

Raise $3$ to the power of $2$ .

$5 - 9 + 9 + 9 - 6 \cdot 4 + 4^{2}$

Step 3.13

Add $- 9$ and $9$ .

$5 + 0 + 9 - 6 \cdot 4 + 4^{2}$

Step 3.14

Add $5$ and $0$ .

$5 + 9 - 6 \cdot 4 + 4^{2}$

Step 3.15

Multiply $- 6$ by $4$ .

$5 + 9 - 24 + 4^{2}$

Step 3.16

Subtract $24$ from $9$ .

$5 - 15 + 4^{2}$

Step 3.17

Raise $4$ to the power of $2$ .

$5 - 15 + 16$

Step 3.18

Add $- 15$ and $16$ .

$5 + 1$

Step 3.19

Add $5$ and $1$ .

$6$