Calculus Examples

$7$ , $3$ , $- 1$ , $- 5$

Step 1

This is the formula to find the sum of the first $n$ terms of the sequence. To evaluate it, the values of the first and $n$ th terms must be found.

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

Step 2

This is an arithmetic sequence since there is a common difference between each term. In this case, adding $- 4$ to the previous term in the sequence gives the next term. In other words, $a_{n} = a_{1} + d (n - 1)$ .

Arithmetic Sequence: $d = - 4$

Step 3

This is the formula of an arithmetic sequence.

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

Step 4

Substitute in the values of $a_{1} = 7$ and $d = - 4$ .

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

Step 5

Simplify each term.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Multiply $- 4$ by $- 1$ .

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

Step 6

Add $7$ and $4$ .

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

Step 7

Substitute in the value of $n$ to find the $n$ th term.

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

Step 8

Multiply $- 4$ by $6$ .

$a_{6} = - 24 + 11$

Step 9

Add $- 24$ and $11$ .

$a_{6} = - 13$

Step 10

Replace the variables with the known values to find $S_{6}$ .

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

Step 11

Divide $6$ by $2$ .

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

Step 12

Subtract $13$ from $7$ .

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

Step 13

Multiply $3$ by $- 6$ .

$S_{6} = - 18$

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