Find the Sum of the Series 30 , 0(1+0.055/2)^(2*5)

$30$ , $0 {(1 + \frac{0.055}{2})}^{2 \cdot 5}$

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 $- 30$ to the previous term in the sequence gives the next term. In other words, $a_{n} = a_{1} + d (n - 1)$ .

Arithmetic Sequence: $d = - 30$

Step 3

This is the formula of an arithmetic sequence.

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

Step 4

Substitute in the values of $a_{1} = 30$ and $d = - 30$ .

$a_{n} = 30 - 30 (n - 1)$

Step 5

Simplify each term.

Apply the distributive property.

$a_{n} = 30 - 30 n - 30 \cdot - 1$

Multiply $- 30$ by $- 1$ .

$a_{n} = 30 - 30 n + 30$

Step 6

Add $30$ and $30$ .

$a_{n} = - 30 n + 60$

Step 7

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

$a_{2} = - 30 \cdot 2 + 60$

Step 8

Multiply $- 30$ by $2$ .

$a_{2} = - 60 + 60$

Step 9

Add $- 60$ and $60$ .

$a_{2} = 0$

Step 10

Replace the variables with the known values to find $S_{2}$ .

$S_{2} = \frac{2}{2} \cdot (30 + 0)$

Step 11

Cancel the common factor of $2$ .

Cancel the common factor.

$S_{2} = \frac{2}{2} \cdot (30 + 0)$

Rewrite the expression.

$S_{2} = 1 \cdot (30 + 0)$

Step 12

Simplify the expression.

Multiply $30 + 0$ by $1$ .

$S_{2} = 30 + 0$

Add $30$ and $0$ .

$S_{2} = 30$

Step 13

Convert the fraction to a decimal.

$S_{2} = 30$

Find the Sum of the Series 30 , 0(1+0.055/2)^(2*5)

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