Calculus Examples

- 1

$- 1$ ,

- 2

$- 2$ ,

- 3

$- 3$ ,

- 4

$- 4$ ,

- 5

$- 5$ ,

- 6

$- 6$

Step 1

This is the formula to find the sum of the first

n

$n$ terms of the sequence. To evaluate it, the values of the first and

n

$n$ th terms must be found.

S_{n} = \frac{n}{2} \cdot (a_{1} + a_{n})

$S_{n} = \frac{n}{2} \cdot (a_{1} + a_{n})$

Step 2

This is an arithmetic sequence since there is a common difference between each term. In this case, adding

- 1

$- 1$ to the previous term in the sequence gives the next term. In other words,

a_{n} = a_{1} + d (n - 1)

$a_{n} = a_{1} + d (n - 1)$ .

Arithmetic Sequence:

d = - 1

$d = - 1$

Step 3

This is the formula of an arithmetic sequence.

a_{n} = a_{1} + d (n - 1)

$a_{n} = a_{1} + d (n - 1)$

Step 4

Substitute in the values of

a_{1} = - 1

$a_{1} = - 1$ and

d = - 1

$d = - 1$ .

a_{n} = - 1 - (n - 1)

$a_{n} = - 1 - (n - 1)$

Step 5

Simplify each term.

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Step 5.1

Apply the distributive property.

a_{n} = - 1 - n - - 1

$a_{n} = - 1 - n - - 1$

Step 5.2

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

a_{n} = - 1 - n + 1

$a_{n} = - 1 - n + 1$

a_{n} = - 1 - n + 1

$a_{n} = - 1 - n + 1$

Step 6

Combine the opposite terms in

- 1 - n + 1

$- 1 - n + 1$ .

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Step 6.1

Add

- 1

$- 1$ and

1

$1$ .

a_{n} = - n + 0

$a_{n} = - n + 0$

Step 6.2

Add

- n

$- n$ and

0

$0$ .

a_{n} = - n

$a_{n} = - n$

a_{n} = - n

$a_{n} = - n$

Step 7

Substitute in the value of

n

$n$ to find the

n

$n$ th term.

a_{6} = - (6)

$a_{6} = - (6)$

Step 8

Multiply

- 1

$- 1$ by

6

$6$ .

a_{6} = - 6

$a_{6} = - 6$

Step 9

Replace the variables with the known values to find

S_{6}

$S_{6}$ .

S_{6} = \frac{6}{2} \cdot (- 1 - 6)

$S_{6} = \frac{6}{2} \cdot (- 1 - 6)$

Step 10

Divide

6

$6$ by

2

$2$ .

S_{6} = 3 \cdot (- 1 - 6)

$S_{6} = 3 \cdot (- 1 - 6)$

Step 11

Subtract

6

$6$ from

- 1

$- 1$ .

S_{6} = 3 \cdot - 7

$S_{6} = 3 \cdot - 7$

Step 12

Multiply

3

$3$ by

- 7

$- 7$ .

S_{6} = - 21

$S_{6} = - 21$

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