Calculus Examples

2

$2$ ,

5

$5$ ,

8

$8$ ,

11

$11$ ,

14

$14$

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

3

$3$ 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 = 3

$d = 3$

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} = 2

$a_{1} = 2$ and

d = 3

$d = 3$ .

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

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

Step 5

Simplify each term.

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Step 5.1

Apply the distributive property.

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

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

Step 5.2

Multiply

3

$3$ by

- 1

$- 1$ .

a_{n} = 2 + 3 n - 3

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

a_{n} = 2 + 3 n - 3

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

Step 6

Subtract

3

$3$ from

2

$2$ .

a_{n} = 3 n - 1

$a_{n} = 3 n - 1$

Step 7

Substitute in the value of

n

$n$ to find the

n

$n$ th term.

a_{7} = 3 (7) - 1

$a_{7} = 3 (7) - 1$

Step 8

Multiply

3

$3$ by

7

$7$ .

a_{7} = 21 - 1

$a_{7} = 21 - 1$

Step 9

Subtract

1

$1$ from

21

$21$ .

a_{7} = 20

$a_{7} = 20$

Step 10

Replace the variables with the known values to find

S_{7}

$S_{7}$ .

S_{7} = \frac{7}{2} \cdot (2 + 20)

$S_{7} = \frac{7}{2} \cdot (2 + 20)$

Step 11

Add

2

$2$ and

20

$20$ .

S_{7} = \frac{7}{2} \cdot 22

$S_{7} = \frac{7}{2} \cdot 22$

Step 12

Cancel the common factor of

2

$2$ .

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Step 12.1

Factor

2

$2$ out of

22

$22$ .

S_{7} = \frac{7}{2} \cdot (2 (11))

$S_{7} = \frac{7}{2} \cdot (2 (11))$

Step 12.2

Cancel the common factor.

S_{7} = \frac{7}{2} \cdot (2 \cdot 11)

$S_{7} = \frac{7}{2} \cdot (2 \cdot 11)$

Step 12.3

Rewrite the expression.

S_{7} = 7 \cdot 11

$S_{7} = 7 \cdot 11$

S_{7} = 7 \cdot 11

$S_{7} = 7 \cdot 11$

Step 13

Multiply

7

$7$ by

11

$11$ .

S_{7} = 77

$S_{7} = 77$

Step 14

Convert the fraction to a decimal.

S_{7} = 77

$S_{7} = 77$

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