Calculus Examples

$\sum_{k = 2}^{4} k (k + 3)$

Step 1

Simplify the summation.

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Step 1.1

Apply the distributive property.

$k \cdot k + k \cdot 3$

Step 1.2

Multiply $k$ by $k$ .

$k^{2} + k \cdot 3$

Step 1.3

Move $3$ to the left of $k$ .

$k^{2} + 3 k$

Step 1.4

Rewrite the summation.

$\sum_{k = 2}^{4} k^{2} + 3 k$

Step 2

Expand the series for each value of $k$ .

$2^{2} + 3 \cdot 2 + 3^{2} + 3 \cdot 3 + 4^{2} + 3 \cdot 4$

Step 3

Simplify.

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Step 3.1

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

$4 + 3 \cdot 2 + 3^{2} + 3 \cdot 3 + 4^{2} + 3 \cdot 4$

Step 3.2

Multiply $3$ by $2$ .

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

Step 3.3

Add $4$ and $6$ .

$10 + 3^{2} + 3 \cdot 3 + 4^{2} + 3 \cdot 4$

Step 3.4

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

$10 + 9 + 3 \cdot 3 + 4^{2} + 3 \cdot 4$

Step 3.5

Multiply $3$ by $3$ .

$10 + 9 + 9 + 4^{2} + 3 \cdot 4$

Step 3.6

Add $9$ and $9$ .

$10 + 18 + 4^{2} + 3 \cdot 4$

Step 3.7

Add $10$ and $18$ .

$28 + 4^{2} + 3 \cdot 4$

Step 3.8

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

$28 + 16 + 3 \cdot 4$

Step 3.9

Multiply $3$ by $4$ .

$28 + 16 + 12$

Step 3.10

Add $16$ and $12$ .

$28 + 28$

Step 3.11

Add $28$ and $28$ .

$56$