Calculus Examples

$\sum_{n = 1}^{4} {(a - n)}^{2}$

Step 1

Simplify the summation.

Tap for more steps...

Step 1.1

Rewrite ${(a - n)}^{2}$ as $(a - n) (a - n)$ .

$(a - n) (a - n)$

Step 1.2

Expand $(a - n) (a - n)$ using the FOIL Method.

Tap for more steps...

Step 1.2.1

Apply the distributive property.

$a (a - n) - n (a - n)$

Step 1.2.2

Apply the distributive property.

$a \cdot a + a (- n) - n (a - n)$

Step 1.2.3

Apply the distributive property.

$a \cdot a + a (- n) - n a - n (- n)$

Step 1.3

Simplify and combine like terms.

Tap for more steps...

Step 1.3.1

Simplify each term.

Tap for more steps...

Step 1.3.1.1

Multiply $a$ by $a$ .

$a^{2} + a (- n) - n a - n (- n)$

Step 1.3.1.2

Rewrite using the commutative property of multiplication.

$a^{2} - a n - n a - n (- n)$

Step 1.3.1.3

Rewrite using the commutative property of multiplication.

$a^{2} - a n - n a - 1 \cdot - 1 n \cdot n$

Step 1.3.1.4

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

Tap for more steps...

Step 1.3.1.4.1

Move $n$ .

$a^{2} - a n - n a - 1 \cdot - 1 (n \cdot n)$

Step 1.3.1.4.2

Multiply $n$ by $n$ .

$a^{2} - a n - n a - 1 \cdot - 1 n^{2}$

Step 1.3.1.5

Multiply $- 1$ by $- 1$ .

$a^{2} - a n - n a + 1 n^{2}$

Step 1.3.1.6

Multiply $n^{2}$ by $1$ .

$a^{2} - a n - n a + n^{2}$

Step 1.3.2

Subtract $n a$ from $- a n$ .

Tap for more steps...

Step 1.3.2.1

Move $n$ .

$a^{2} - a n - 1 a n + n^{2}$

Step 1.3.2.2

Subtract $a n$ from $- a n$ .

$a^{2} - 2 a n + n^{2}$

Step 1.4

Rewrite the summation.

$\sum_{n = 1}^{4} a^{2} - 2 a n + n^{2}$

Step 2

Expand the series for each value of $n$ .

$a^{2} - 2 a \cdot 1 + 1^{2} + a^{2} - 2 a \cdot 2 + 2^{2} + a^{2} - 2 a \cdot 3 + 3^{2} + a^{2} - 2 a \cdot 4 + 4^{2}$

Step 3

Substitute the values back and simplify $a^{2} - 2 a \cdot 1 + 1^{2} + a^{2} - 2 a \cdot 2 + 2^{2} + a^{2} - 2 a \cdot 3 + 3^{2} + a^{2} - 2 a \cdot 4 + 4^{2}$ .

Tap for more steps...

Step 3.1

Simplify each term.

Tap for more steps...

Step 3.1.1

Multiply $- 2$ by $1$ .

$a^{2} - 2 a + 1^{2} + a^{2} - 2 a \cdot 2 + 2^{2} + a^{2} - 2 a \cdot 3 + 3^{2} + a^{2} - 2 a \cdot 4 + 4^{2}$

Step 3.1.2

One to any power is one.

$a^{2} - 2 a + 1 + a^{2} - 2 a \cdot 2 + 2^{2} + a^{2} - 2 a \cdot 3 + 3^{2} + a^{2} - 2 a \cdot 4 + 4^{2}$

Step 3.1.3

Multiply $2$ by $- 2$ .

$a^{2} - 2 a + 1 + a^{2} - 4 a + 2^{2} + a^{2} - 2 a \cdot 3 + 3^{2} + a^{2} - 2 a \cdot 4 + 4^{2}$

Step 3.1.4

Raise $2$ to the power of $2$ .

$a^{2} - 2 a + 1 + a^{2} - 4 a + 4 + a^{2} - 2 a \cdot 3 + 3^{2} + a^{2} - 2 a \cdot 4 + 4^{2}$

Step 3.1.5

Multiply $3$ by $- 2$ .

$a^{2} - 2 a + 1 + a^{2} - 4 a + 4 + a^{2} - 6 a + 3^{2} + a^{2} - 2 a \cdot 4 + 4^{2}$

Step 3.1.6

Raise $3$ to the power of $2$ .

$a^{2} - 2 a + 1 + a^{2} - 4 a + 4 + a^{2} - 6 a + 9 + a^{2} - 2 a \cdot 4 + 4^{2}$

Step 3.1.7

Multiply $4$ by $- 2$ .

$a^{2} - 2 a + 1 + a^{2} - 4 a + 4 + a^{2} - 6 a + 9 + a^{2} - 8 a + 4^{2}$

Step 3.1.8

Raise $4$ to the power of $2$ .

$a^{2} - 2 a + 1 + a^{2} - 4 a + 4 + a^{2} - 6 a + 9 + a^{2} - 8 a + 16$

Step 3.2

Simplify by adding terms.

Tap for more steps...

Step 3.2.1

Add $a^{2}$ and $a^{2}$ .

$2 a^{2} - 2 a + 1 - 4 a + 4 + a^{2} - 6 a + 9 + a^{2} - 8 a + 16$

Step 3.2.2

Add $2 a^{2}$ and $a^{2}$ .

$3 a^{2} - 2 a + 1 - 4 a + 4 - 6 a + 9 + a^{2} - 8 a + 16$

Step 3.2.3

Add $3 a^{2}$ and $a^{2}$ .

$4 a^{2} - 2 a + 1 - 4 a + 4 - 6 a + 9 - 8 a + 16$

Step 3.2.4

Subtract $4 a$ from $- 2 a$ .

$4 a^{2} - 6 a + 1 + 4 - 6 a + 9 - 8 a + 16$

Step 3.2.5

Subtract $6 a$ from $- 6 a$ .

$4 a^{2} - 12 a + 1 + 4 + 9 - 8 a + 16$

Step 3.2.6

Subtract $8 a$ from $- 12 a$ .

$4 a^{2} - 20 a + 1 + 4 + 9 + 16$

Step 3.2.7

Simplify by adding numbers.

Tap for more steps...

Step 3.2.7.1

Add $1$ and $4$ .

$4 a^{2} - 20 a + 5 + 9 + 16$

Step 3.2.7.2

Add $5$ and $9$ .

$4 a^{2} - 20 a + 14 + 16$

Step 3.2.7.3

Add $14$ and $16$ .

$4 a^{2} - 20 a + 30$