Algebra Examples

$2$ ,

$4$ ,

$6$ ,

$8$ ,

$10$

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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Step 5.1

Apply the distributive property.

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

Step 5.2

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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Step 6.1

Subtract

$2$ from

$2$ .

$a_{n} = 2 n + 0$

Step 6.2

Add

$2 n$ and

$0$ .

$a_{n} = 2 n$

Step 7

Substitute in the value of

$n$ to find the

$n$ th term.

$a_{5} = 2 (5)$

Step 8

Multiply

$2$ by

$5$ .

$a_{5} = 10$

Step 9

Replace the variables with the known values to find

$S_{5}$ .

$S_{5} = \frac{5}{2} \cdot (2 + 10)$

Step 10

Add

$2$ and

$10$ .

$S_{5} = \frac{5}{2} \cdot 12$

Step 11

Cancel the common factor of

$2$ .

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Step 11.1

Factor

$2$ out of

$12$ .

$S_{5} = \frac{5}{2} \cdot (2 (6))$

Step 11.2

Cancel the common factor.

$S_{5} = \frac{5}{2} \cdot (2 \cdot 6)$

Step 11.3

Rewrite the expression.

$S_{5} = 5 \cdot 6$

Step 12

Multiply

$5$ by

$6$ .

$S_{5} = 30$

Step 13

Convert the fraction to a decimal.

$S_{5} = 30$