Algebra Examples

$2 + 4 + 6 + 8$

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 $2$ to the previous term in the sequence gives the next term. In other words, $a_{n} = a_{1} + d (n - 1)$ .

Arithmetic Sequence: $d = 2$

Step 3

This is the formula of an arithmetic sequence.

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

Step 4

Substitute in the values of $a_{1} = 2$ and $d = 2$ .

$a_{n} = 2 + 2 (n - 1)$

Step 5

Simplify each term.

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Apply the distributive property.

$a_{n} = 2 + 2 n + 2 \cdot - 1$

Multiply $2$ by $- 1$ .

$a_{n} = 2 + 2 n - 2$

Step 6

Combine the opposite terms in $2 + 2 n - 2$ .

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Subtract $2$ from $2$ .

$a_{n} = 2 n + 0$

Add $2 n$ and $0$ .

$a_{n} = 2 n$

Step 7

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

$a_{4} = 2 (4)$

Step 8

Multiply $2$ by $4$ .

$a_{4} = 8$

Step 9

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

$S_{4} = \frac{4}{2} \cdot (2 + 8)$

Step 10

Divide $4$ by $2$ .

$S_{4} = 2 \cdot (2 + 8)$

Step 11

Add $2$ and $8$ .

$S_{4} = 2 \cdot 10$

Step 12

Multiply $2$ by $10$ .

$S_{4} = 20$

Step 13

Convert the fraction to a decimal.

$S_{4} = 20$