Calculus Examples

$\sum_{j = 0}^{3} 3 - j n$

Step 1

Expand the series for each value of $j$ .

$3 + 0 n + 3 - 1 n + 3 - 2 n + 3 - 3 n$

Step 2

Substitute the values back and simplify $3 + 0 n + 3 - 1 n + 3 - 2 n + 3 - 3 n$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Multiply $0$ by $n$ .

$3 + 0 + 3 - 1 n + 3 - 2 n + 3 - 3 n$

Step 2.1.2

Rewrite $- 1 n$ as $- n$ .

$3 + 0 + 3 - n + 3 - 2 n + 3 - 3 n$

Step 2.2

Simplify by adding terms.

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Step 2.2.1

Add $3$ and $0$ .

$3 + 3 - n + 3 - 2 n + 3 - 3 n$

Step 2.2.2

Simplify by adding numbers.

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Step 2.2.2.1

Add $3$ and $3$ .

$6 - n + 3 - 2 n + 3 - 3 n$

Step 2.2.2.2

Add $6$ and $3$ .

$- n + 9 - 2 n + 3 - 3 n$

Step 2.2.3

Subtract $2 n$ from $- n$ .

$- 3 n + 9 + 3 - 3 n$

Step 2.2.4

Subtract $3 n$ from $- 3 n$ .

$- 6 n + 9 + 3$

Step 2.2.5

Add $9$ and $3$ .

$- 6 n + 12$