Precalculus Examples

6

$6$ ,

8

$8$

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

2

$2$ 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 = 2

$d = 2$

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} = 6

$a_{1} = 6$ and

d = 2

$d = 2$ .

a_{n} = 6 + 2 (n - 1)

$a_{n} = 6 + 2 (n - 1)$

Step 5

Simplify each term.

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Step 5.1

Apply the distributive property.

a_{n} = 6 + 2 n + 2 \cdot - 1

$a_{n} = 6 + 2 n + 2 \cdot - 1$

Step 5.2

Multiply

2

$2$ by

- 1

$- 1$ .

a_{n} = 6 + 2 n - 2

$a_{n} = 6 + 2 n - 2$

a_{n} = 6 + 2 n - 2

$a_{n} = 6 + 2 n - 2$

Step 6

Subtract

2

$2$ from

6

$6$ .

a_{n} = 2 n + 4

$a_{n} = 2 n + 4$

Step 7

Substitute in the value of

n

$n$ to find the

n

$n$ th term.

a_{2} = 2 (2) + 4

$a_{2} = 2 (2) + 4$

Step 8

Multiply

2

$2$ by

2

$2$ .

a_{2} = 4 + 4

$a_{2} = 4 + 4$

Step 9

Add

4

$4$ and

4

$4$ .

a_{2} = 8

$a_{2} = 8$

Step 10

Replace the variables with the known values to find

S_{2}

$S_{2}$ .

S_{2} = \frac{2}{2} \cdot (6 + 8)

$S_{2} = \frac{2}{2} \cdot (6 + 8)$

Step 11

Cancel the common factor of

2

$2$ .

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Step 11.1

Cancel the common factor.

S_{2} = \frac{2}{2} \cdot (6 + 8)

$S_{2} = \frac{2}{2} \cdot (6 + 8)$

Step 11.2

Rewrite the expression.

S_{2} = 1 \cdot (6 + 8)

$S_{2} = 1 \cdot (6 + 8)$

S_{2} = 1 \cdot (6 + 8)

$S_{2} = 1 \cdot (6 + 8)$

Step 12

Simplify the expression.

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Step 12.1

Multiply

6 + 8

$6 + 8$ by

1

$1$ .

S_{2} = 6 + 8

$S_{2} = 6 + 8$

Step 12.2

Add

6

$6$ and

8

$8$ .

S_{2} = 14

$S_{2} = 14$

S_{2} = 14

$S_{2} = 14$

Step 13

Convert the fraction to a decimal.

S_{2} = 14

$S_{2} = 14$

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