Calculus Examples

$2$ , $4$ , $6$

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.

$a_{3} = 2 (3)$

Step 8

Multiply $2$ by $3$ .

$a_{3} = 6$

Step 9

Replace the variables with the known values to find $S_{3}$ .

$S_{3} = \frac{3}{2} \cdot (2 + 6)$

Step 10

Add $2$ and $6$ .

$S_{3} = \frac{3}{2} \cdot 8$

Step 11

Cancel the common factor of $2$ .

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Step 11.1

Factor $2$ out of $8$ .

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

Step 11.2

Cancel the common factor.

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

Step 11.3

Rewrite the expression.

$S_{3} = 3 \cdot 4$

Step 12

Multiply $3$ by $4$ .

$S_{3} = 12$

Step 13

Convert the fraction to a decimal.

$S_{3} = 12$

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