Calculus Examples

$7 + 14 + 21 + 28 + 35 + 42$

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 $7$ to the previous term in the sequence gives the next term. In other words, $a_{n} = a_{1} + d (n - 1)$ .

Arithmetic Sequence: $d = 7$

Step 3

This is the formula of an arithmetic sequence.

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

Step 4

Substitute in the values of $a_{1} = 7$ and $d = 7$ .

$a_{n} = 7 + 7 (n - 1)$

Step 5

Simplify each term.

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Step 5.1

Apply the distributive property.

$a_{n} = 7 + 7 n + 7 \cdot - 1$

Step 5.2

Multiply $7$ by $- 1$ .

$a_{n} = 7 + 7 n - 7$

Step 6

Combine the opposite terms in $7 + 7 n - 7$ .

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Step 6.1

Subtract $7$ from $7$ .

$a_{n} = 7 n + 0$

Step 6.2

Add $7 n$ and $0$ .

$a_{n} = 7 n$

Step 7

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

$a_{6} = 7 (6)$

Step 8

Multiply $7$ by $6$ .

$a_{6} = 42$

Step 9

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

$S_{6} = \frac{6}{2} \cdot (7 + 42)$

Step 10

Divide $6$ by $2$ .

$S_{6} = 3 \cdot (7 + 42)$

Step 11

Add $7$ and $42$ .

$S_{6} = 3 \cdot 49$

Step 12

Multiply $3$ by $49$ .

$S_{6} = 147$

Step 13

Convert the fraction to a decimal.

$S_{6} = 147$