Precalculus Examples

$22$ , $19$ , $16$ , $13$

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 $- 3$ to the previous term in the sequence gives the next term. In other words, $a_{n} = a_{1} + d (n - 1)$ .

Arithmetic Sequence: $d = - 3$

Step 3

This is the formula of an arithmetic sequence.

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

Step 4

Substitute in the values of $a_{1} = 22$ and $d = - 3$ .

$a_{n} = 22 - 3 (n - 1)$

Step 5

Simplify each term.

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Step 5.1

Apply the distributive property.

$a_{n} = 22 - 3 n - 3 \cdot - 1$

Step 5.2

Multiply $- 3$ by $- 1$ .

$a_{n} = 22 - 3 n + 3$

Step 6

Add $22$ and $3$ .

$a_{n} = - 3 n + 25$

Step 7

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

$a_{4} = - 3 \cdot 4 + 25$

Step 8

Multiply $- 3$ by $4$ .

$a_{4} = - 12 + 25$

Step 9

Add $- 12$ and $25$ .

$a_{4} = 13$

Step 10

Replace the variables with the known values to find $S_{4}$ .

$S_{4} = \frac{4}{2} \cdot (22 + 13)$

Step 11

Divide $4$ by $2$ .

$S_{4} = 2 \cdot (22 + 13)$

Step 12

Add $22$ and $13$ .

$S_{4} = 2 \cdot 35$

Step 13

Multiply $2$ by $35$ .

$S_{4} = 70$

Step 14

Convert the fraction to a decimal.

$S_{4} = 70$