Calculus Examples

0.1

$0.1$ ,

0.2

$0.2$ ,

0.3

$0.3$ ,

0.4

$0.4$ ,

0.5

$0.5$ ,

0.6

$0.6$ ,

0.7

$0.7$ ,

0.8

$0.8$ ,

0.9

$0.9$

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

0.1

$0.1$ 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 = 0.1

$d = 0.1$

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} = 0.1

$a_{1} = 0.1$ and

d = 0.1

$d = 0.1$ .

a_{n} = 0.1 + 0.1 (n - 1)

$a_{n} = 0.1 + 0.1 (n - 1)$

Step 5

Simplify each term.

Tap for more steps...

Step 5.1

Apply the distributive property.

a_{n} = 0.1 + 0.1 n + 0.1 \cdot - 1

$a_{n} = 0.1 + 0.1 n + 0.1 \cdot - 1$

Step 5.2

Multiply

0.1

$0.1$ by

- 1

$- 1$ .

a_{n} = 0.1 + 0.1 n - 0.1

$a_{n} = 0.1 + 0.1 n - 0.1$

a_{n} = 0.1 + 0.1 n - 0.1

$a_{n} = 0.1 + 0.1 n - 0.1$

Step 6

Combine the opposite terms in

0.1 + 0.1 n - 0.1

$0.1 + 0.1 n - 0.1$ .

Tap for more steps...

Step 6.1

Subtract

0.1

$0.1$ from

0.1

$0.1$ .

a_{n} = 0.1 n + 0

$a_{n} = 0.1 n + 0$

Step 6.2

Add

0.1 n

$0.1 n$ and

0

$0$ .

a_{n} = 0.1 n

$a_{n} = 0.1 n$

a_{n} = 0.1 n

$a_{n} = 0.1 n$

Step 7

Substitute in the value of

n

$n$ to find the

n

$n$ th term.

a_{9} = 0.1 (9)

$a_{9} = 0.1 (9)$

Step 8

Multiply

0.1

$0.1$ by

9

$9$ .

a_{9} = 0.9

$a_{9} = 0.9$

Step 9

Replace the variables with the known values to find

S_{9}

$S_{9}$ .

S_{9} = \frac{9}{2} \cdot (0.1 + 0.9)

$S_{9} = \frac{9}{2} \cdot (0.1 + 0.9)$

Step 10

Add

0.1

$0.1$ and

0.9

$0.9$ .

S_{9} = \frac{9}{2} \cdot 1

$S_{9} = \frac{9}{2} \cdot 1$

Step 11

Cancel the common factor of

1

$1$ .

Tap for more steps...

Step 11.1

Rewrite

2

$2$ as

1 (2)

$1 (2)$ .

S_{9} = \frac{9}{1 (2)} \cdot 1

$S_{9} = \frac{9}{1 (2)} \cdot 1$

Step 11.2

Cancel the common factor.

$S_{9} = \frac{9}{1 \cdot 2} \cdot 1$

Step 11.3

Rewrite the expression.

$S_{9} = \frac{9}{2}$

Step 12

Convert the fraction to a decimal.

$S_{9} = 4.5$