Algebra Examples

3 + 6 + 9 + 12

$3 + 6 + 9 + 12$

Step 1

This is the formula to find the sum of the first

n

$n$ terms of the sequence. To evaluate it, the values of the first and

n

$n$ th terms must be found.

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

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

Step 2

This is an arithmetic sequence since there is a common difference between each term. In this case, adding

3

$3$ to the previous term in the sequence gives the next term. In other words,

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

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

Arithmetic Sequence:

d = 3

$d = 3$

Step 3

This is the formula of an arithmetic sequence.

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

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

Step 4

Substitute in the values of

a_{1} = 3

$a_{1} = 3$ and

d = 3

$d = 3$ .

a_{n} = 3 + 3 (n - 1)

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

Step 5

Simplify each term.

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Step 5.1

Apply the distributive property.

a_{n} = 3 + 3 n + 3 \cdot - 1

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

Step 5.2

Multiply

3

$3$ by

- 1

$- 1$ .

a_{n} = 3 + 3 n - 3

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

a_{n} = 3 + 3 n - 3

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

Step 6

Combine the opposite terms in

3 + 3 n - 3

$3 + 3 n - 3$ .

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Step 6.1

Subtract

3

$3$ from

3

$3$ .

a_{n} = 3 n + 0

$a_{n} = 3 n + 0$

Step 6.2

Add

3 n

$3 n$ and

0

$0$ .

a_{n} = 3 n

$a_{n} = 3 n$

a_{n} = 3 n

$a_{n} = 3 n$

Step 7

Substitute in the value of

n

$n$ to find the

n

$n$ th term.

a_{4} = 3 (4)

$a_{4} = 3 (4)$

Step 8

Multiply

3

$3$ by

4

$4$ .

a_{4} = 12

$a_{4} = 12$

Step 9

Replace the variables with the known values to find

S_{4}

$S_{4}$ .

S_{4} = \frac{4}{2} \cdot (3 + 12)

$S_{4} = \frac{4}{2} \cdot (3 + 12)$

Step 10

Divide

4

$4$ by

2

$2$ .

S_{4} = 2 \cdot (3 + 12)

$S_{4} = 2 \cdot (3 + 12)$

Step 11

Add

3

$3$ and

12

$12$ .

S_{4} = 2 \cdot 15

$S_{4} = 2 \cdot 15$

Step 12

Multiply

2

$2$ by

15

$15$ .

S_{4} = 30

$S_{4} = 30$

Step 13

Convert the fraction to a decimal.

S_{4} = 30

$S_{4} = 30$