Calculus Examples

$\sum_{i = 8}^{6} {(i - 1)}^{2}$

Step 1

Simplify the summation.

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Step 1.1

Rewrite ${(i - 1)}^{2}$ as $(i - 1) (i - 1)$ .

$(i - 1) (i - 1)$

Step 1.2

Expand $(i - 1) (i - 1)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$i (i - 1) - 1 (i - 1)$

Step 1.2.2

Apply the distributive property.

$i \cdot i + i \cdot - 1 - 1 (i - 1)$

Step 1.2.3

Apply the distributive property.

$i \cdot i + i \cdot - 1 - 1 i - 1 \cdot - 1$

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Multiply $i$ by $i$ .

$i^{2} + i \cdot - 1 - 1 i - 1 \cdot - 1$

Step 1.3.1.2

Move $- 1$ to the left of $i$ .

$i^{2} - 1 \cdot i - 1 i - 1 \cdot - 1$

Step 1.3.1.3

Rewrite $- 1 i$ as $- i$ .

$i^{2} - i - 1 i - 1 \cdot - 1$

Step 1.3.1.4

Rewrite $- 1 i$ as $- i$ .

$i^{2} - i - i - 1 \cdot - 1$

Step 1.3.1.5

Multiply $- 1$ by $- 1$ .

$i^{2} - i - i + 1$

Step 1.3.2

Subtract $i$ from $- i$ .

$i^{2} - 2 i + 1$

Step 1.4

Rewrite the summation.

$\sum_{i = 8}^{6} i^{2} - 2 i + 1$

Step 2

Since the summation's lower limit is greater than the summation's upper limit, it is an empty summation and its value is defined to be zero.

$0$