Precalculus Examples

$22 + 27 + 32 + 37 + 42 + 47$

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 $5$ to the previous term in the sequence gives the next term. In other words, $a_{n} = a_{1} + d (n - 1)$ .

Arithmetic Sequence: $d = 5$

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 $d = 5$ .

$a_{n} = 22 + 5 (n - 1)$

Step 5

Simplify each term.

Tap for more steps...

Step 5.1

Apply the distributive property.

$a_{n} = 22 + 5 n + 5 \cdot - 1$

Step 5.2

Multiply $5$ by $- 1$ .

$a_{n} = 22 + 5 n - 5$

Step 6

Subtract $5$ from $22$ .

$a_{n} = 5 n + 17$

Step 7

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

$a_{6} = 5 (6) + 17$

Step 8

Multiply $5$ by $6$ .

$a_{6} = 30 + 17$

Step 9

Add $30$ and $17$ .

$a_{6} = 47$

Step 10

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

$S_{6} = \frac{6}{2} \cdot (22 + 47)$

Step 11

Divide $6$ by $2$ .

$S_{6} = 3 \cdot (22 + 47)$

Step 12

Add $22$ and $47$ .

$S_{6} = 3 \cdot 69$

Step 13

Multiply $3$ by $69$ .

$S_{6} = 207$

Step 14

Convert the fraction to a decimal.

$S_{6} = 207$