Calculus Examples

7

$7$ ,

3

$3$ ,

- 1

$- 1$ ,

- 5

$- 5$

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

- 4

$- 4$ 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 = - 4

$d = - 4$

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} = 7

$a_{1} = 7$ and

d = - 4

$d = - 4$ .

a_{n} = 7 - 4 (n - 1)

$a_{n} = 7 - 4 (n - 1)$

Step 5

Simplify each term.

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Step 5.1

Apply the distributive property.

a_{n} = 7 - 4 n - 4 \cdot - 1

$a_{n} = 7 - 4 n - 4 \cdot - 1$

Step 5.2

Multiply

- 4

$- 4$ by

- 1

$- 1$ .

a_{n} = 7 - 4 n + 4

$a_{n} = 7 - 4 n + 4$

a_{n} = 7 - 4 n + 4

$a_{n} = 7 - 4 n + 4$

Step 6

Add

7

$7$ and

4

$4$ .

a_{n} = - 4 n + 11

$a_{n} = - 4 n + 11$

Step 7

Substitute in the value of

n

$n$ to find the

n

$n$ th term.

a_{6} = - 4 \cdot 6 + 11

$a_{6} = - 4 \cdot 6 + 11$

Step 8

Multiply

- 4

$- 4$ by

6

$6$ .

a_{6} = - 24 + 11

$a_{6} = - 24 + 11$

Step 9

Add

- 24

$- 24$ and

11

$11$ .

a_{6} = - 13

$a_{6} = - 13$

Step 10

Replace the variables with the known values to find

S_{6}

$S_{6}$ .

S_{6} = \frac{6}{2} \cdot (7 - 13)

$S_{6} = \frac{6}{2} \cdot (7 - 13)$

Step 11

Divide

6

$6$ by

2

$2$ .

S_{6} = 3 \cdot (7 - 13)

$S_{6} = 3 \cdot (7 - 13)$

Step 12

Subtract

13

$13$ from

7

$7$ .

S_{6} = 3 \cdot - 6

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

Step 13

Multiply

3

$3$ by

- 6

$- 6$ .

S_{6} = - 18

$S_{6} = - 18$

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