Calculus Examples

$3$ , $5$ , $7$ , $9$ , $11$ , $13$

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 $a_{1} = 3$ and $d = 2$ .

$a_{n} = 3 + 2 (n - 1)$

Step 5

Simplify each term.

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Step 5.1

Apply the distributive property.

$a_{n} = 3 + 2 n + 2 \cdot - 1$

Step 5.2

Multiply $2$ by $- 1$ .

$a_{n} = 3 + 2 n - 2$

Step 6

Subtract $2$ from $3$ .

$a_{n} = 2 n + 1$

Step 7

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

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

Step 8

Multiply $2$ by $6$ .

$a_{6} = 12 + 1$

Step 9

Add $12$ and $1$ .

$a_{6} = 13$

Step 10

Replace the variables with the known values to find $S_{6}$ .

$S_{6} = \frac{6}{2} \cdot (3 + 13)$

Step 11

Divide $6$ by $2$ .

$S_{6} = 3 \cdot (3 + 13)$

Step 12

Add $3$ and $13$ .

$S_{6} = 3 \cdot 16$

Step 13

Multiply $3$ by $16$ .

$S_{6} = 48$

Step 14

Convert the fraction to a decimal.

$S_{6} = 48$

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