Calculus Examples

$\sum_{j = 1}^{200} 2 j (j + 3)$

Step 1

Simplify the summation.

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Step 1.1

Apply the distributive property.

$2 j \cdot j + 2 j \cdot 3$

Step 1.2

Multiply $j$ by $j$ by adding the exponents.

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Step 1.2.1

Move $j$ .

$2 (j \cdot j) + 2 j \cdot 3$

Step 1.2.2

Multiply $j$ by $j$ .

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

Step 1.3

Multiply $3$ by $2$ .

$2 j^{2} + 6 j$

Step 1.4

Rewrite the summation.

$\sum_{j = 1}^{200} 2 j^{2} + 6 j$

Step 2

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

$\sum_{j = 1}^{200} 2 j^{2} + 6 j = 2 \sum_{j = 1}^{200} j^{2} + 6 \sum_{j = 1}^{200} j$

Step 3

Evaluate $2 \sum_{j = 1}^{200} j^{2}$ .

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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 and make sure to multiply by the front term.

$(2) (\frac{200 (200 + 1) (2 \cdot 200 + 1)}{6})$

Step 3.3

Simplify.

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Step 3.3.1

Simplify the numerator.

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Step 3.3.1.1

Add $200$ and $1$ .

$2 \frac{200 \cdot 201 (2 \cdot 200 + 1)}{6}$

Step 3.3.1.2

Combine exponents.

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Step 3.3.1.2.1

Multiply $200$ by $201$ .

$2 \frac{40200 (2 \cdot 200 + 1)}{6}$

Step 3.3.1.2.2

Multiply $2$ by $200$ .

$2 \frac{40200 (400 + 1)}{6}$

Step 3.3.1.3

Add $400$ and $1$ .

$2 \frac{40200 \cdot 401}{6}$

Step 3.3.2

Reduce the expression by cancelling the common factors.

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Step 3.3.2.1

Multiply $40200$ by $401$ .

$2 (\frac{16120200}{6})$

Step 3.3.2.2

Cancel the common factor of $2$ .

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Step 3.3.2.2.1

Factor $2$ out of $6$ .

$2 \frac{16120200}{2 (3)}$

Step 3.3.2.2.2

Cancel the common factor.

$2 \frac{16120200}{2 \cdot 3}$

Step 3.3.2.2.3

Rewrite the expression.

$\frac{16120200}{3}$

Step 3.3.2.3

Divide $16120200$ by $3$ .

$5373400$

Step 4

Evaluate $6 \sum_{j = 1}^{200} j$ .

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Step 4.1

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

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

Step 4.2

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

$(6) (\frac{200 (200 + 1)}{2})$

Step 4.3

Simplify.

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Step 4.3.1

Simplify the expression.

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Step 4.3.1.1

Add $200$ and $1$ .

$6 \frac{200 \cdot 201}{2}$

Step 4.3.1.2

Multiply $200$ by $201$ .

$6 (\frac{40200}{2})$

Step 4.3.2

Cancel the common factor of $2$ .

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Step 4.3.2.1

Factor $2$ out of $6$ .

$2 (3) \frac{40200}{2}$

Step 4.3.2.2

Cancel the common factor.

$2 \cdot 3 \frac{40200}{2}$

Step 4.3.2.3

Rewrite the expression.

$3 \cdot 40200$

Step 4.3.3

Multiply $3$ by $40200$ .

$120600$

Step 5

Add the results of the summations.

$5373400 + 120600$

Step 6

Add $5373400$ and $120600$ .

$5494000$