Calculus Examples

$\sum_{n = 1}^{50} n (4 n + 3)$

Step 1

Simplify the summation.

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Apply the distributive property.

$n (4 n) + n \cdot 3$

Rewrite using the commutative property of multiplication.

$4 n \cdot n + n \cdot 3$

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

$4 n \cdot n + 3 \cdot n$

Multiply $n$ by $n$ by adding the exponents.

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Move $n$ .

$4 (n \cdot n) + 3 \cdot n$

Multiply $n$ by $n$ .

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

$4 n^{2} + 3 n$

Rewrite the summation.

$\sum_{n = 1}^{50} 4 n^{2} + 3 n$

Step 2

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

$\sum_{n = 1}^{50} 4 n^{2} + 3 n = 4 \sum_{n = 1}^{50} n^{2} + 3 \sum_{n = 1}^{50} n$

Step 3

Evaluate $4 \sum_{n = 1}^{50} n^{2}$ .

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The formula for the summation of a polynomial with degree $2$ is:

$\sum_{k = 1}^{n} k^{2} = \frac{n (n + 1) (2 n + 1)}{6}$

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

$(4) (\frac{50 (50 + 1) (2 \cdot 50 + 1)}{6})$

Simplify.

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Simplify the numerator.

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Add $50$ and $1$ .

$4 \frac{50 \cdot 51 (2 \cdot 50 + 1)}{6}$

Combine exponents.

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Multiply $50$ by $51$ .

$4 \frac{2550 (2 \cdot 50 + 1)}{6}$

Multiply $2$ by $50$ .

$4 \frac{2550 (100 + 1)}{6}$

Add $100$ and $1$ .

$4 \frac{2550 \cdot 101}{6}$

Simplify terms.

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Multiply $2550$ by $101$ .

$4 (\frac{257550}{6})$

Cancel the common factor of $2$ .

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Factor $2$ out of $4$ .

$2 (2) \frac{257550}{6}$

Factor $2$ out of $6$ .

$2 \cdot 2 \frac{257550}{2 \cdot 3}$

Cancel the common factor.

$2 \cdot 2 \frac{257550}{2 \cdot 3}$

Rewrite the expression.

$2 (\frac{257550}{3})$

Combine $2$ and $\frac{257550}{3}$ .

$\frac{2 \cdot 257550}{3}$

Simplify the expression.

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Multiply $2$ by $257550$ .

$\frac{515100}{3}$

Divide $515100$ by $3$ .

$171700$

Step 4

Evaluate $3 \sum_{n = 1}^{50} n$ .

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

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

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

$(3) (\frac{50 (50 + 1)}{2})$

Simplify.

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Add $50$ and $1$ .

$3 \frac{50 \cdot 51}{2}$

Multiply $50$ by $51$ .

$3 (\frac{2550}{2})$

Divide $2550$ by $2$ .

$3 \cdot 1275$

Multiply $3$ by $1275$ .

$3825$

Step 5

Add the results of the summations.

$171700 + 3825$

Step 6

Add $171700$ and $3825$ .

$175525$