Calculus Examples

$\sum_{n = 51}^{100} 5 n$

Step 1

Split the summation to make the starting value of $n$ equal to $1$ .

$\sum_{n = 51}^{100} 5 n = \sum_{n = 1}^{100} 5 n - \sum_{n = 1}^{50} 5 n$

Step 2

Evaluate $\sum_{n = 1}^{100} 5 n$ .

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Step 2.1

Factor $5$ out of the summation.

$(5) \sum_{n = 1}^{100} n$

Step 2.2

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

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

Step 2.3

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

$(5) (\frac{100 (100 + 1)}{2})$

Step 2.4

Simplify.

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Step 2.4.1

Add $100$ and $1$ .

$5 \frac{100 \cdot 101}{2}$

Step 2.4.2

Multiply $100$ by $101$ .

$5 (\frac{10100}{2})$

Step 2.4.3

Divide $10100$ by $2$ .

$5 \cdot 5050$

Step 2.4.4

Multiply $5$ by $5050$ .

$25250$

Step 3

Evaluate $\sum_{n = 1}^{50} 5 n$ .

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Step 3.1

Factor $5$ out of the summation.

$(5) \sum_{n = 1}^{50} n$

Step 3.2

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

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

Step 3.3

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

$(5) (\frac{50 (50 + 1)}{2})$

Step 3.4

Simplify.

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Step 3.4.1

Add $50$ and $1$ .

$5 \frac{50 \cdot 51}{2}$

Step 3.4.2

Multiply $50$ by $51$ .

$5 (\frac{2550}{2})$

Step 3.4.3

Divide $2550$ by $2$ .

$5 \cdot 1275$

Step 3.4.4

Multiply $5$ by $1275$ .

$6375$

Step 4

Replace the summations with the values found.

$25250 - 6375$

Step 5

Subtract $6375$ from $25250$ .

$18875$