Calculus Examples

$1$ , $3$ , $5$ , $7$ , $9$

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

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

Step 5

Simplify each term.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Multiply $2$ by $- 1$ .

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

Step 6

Subtract $2$ from $1$ .

$a_{n} = 2 n - 1$

Step 7

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

$a_{7} = 2 (7) - 1$

Step 8

Multiply $2$ by $7$ .

$a_{7} = 14 - 1$

Step 9

Subtract $1$ from $14$ .

$a_{7} = 13$

Step 10

Replace the variables with the known values to find $S_{7}$ .

$S_{7} = \frac{7}{2} \cdot (1 + 13)$

Step 11

Add $1$ and $13$ .

$S_{7} = \frac{7}{2} \cdot 14$

Step 12

Cancel the common factor of $2$ .

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Step 12.1

Factor $2$ out of $14$ .

$S_{7} = \frac{7}{2} \cdot (2 (7))$

Step 12.2

Cancel the common factor.

$S_{7} = \frac{7}{2} \cdot (2 \cdot 7)$

Step 12.3

Rewrite the expression.

$S_{7} = 7 \cdot 7$

Step 13

Multiply $7$ by $7$ .

$S_{7} = 49$

Step 14

Convert the fraction to a decimal.

$S_{7} = 49$

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