Problem 53 - Algebra for College Students

Question

Find the sum of each infinite geometric series. $$ \sum_{i=1}^{\infty} 7(0.4)^{i-1} $$

Step-by-Step Solution

Answer

The sum of the infinite geometric series is \(\frac{35}{3}\).

Step 1: Identify the first term (a)

The first term of the series can be found by substituting the first value of the series. In this case, replace \(i\) with 1 in the given series: \(a = 7(0.4)^{1-1} = 7(0.4)^0 = 7(1) = 7\)

Step 2: Identify the common ratio (r)

The common ratio is the factor multiplied to get from one term to the next in the series. Here, the common ratio \(r\) is 0.4, as it is given in the series.

Step 3: Use the formula for the sum of an infinite geometric series

The formula for the sum of an infinite geometric series is \(S = \frac{a}{1-r}\), where \(a\) is the first term and \(r\) is the common ratio.

Step 4: Substitute the known values into the formula

Using the formula \(S = \frac{a}{1-r}\), substitute \(a = 7\) and \(r = 0.4\): \(S = \frac{7}{1-0.4} = \frac{7}{0.6}\)

Step 5: Simplify the expression

Now simplify the expression: \(S = \frac{7}{0.6} = \frac{7 \times 10}{0.6 \times 10} = \frac{70}{6} = \frac{35}{3}\).

Key Concepts

geometric series

A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous term by a constant value, called the common ratio. For example, in the series 7, 2.8, 1.12, ... each term is obtained by multiplying the previous term by 0.4. A geometric series can be finite or infinite. If it's infinite, it continues forever. Geometric series have many applications in fields like finance, physics, and computer science.

series sum formula

The formula to find the sum of an infinite geometric series is very useful. It's given by
\( S = \frac{a}{1-r} \)
where:

• \textbf{a}: the first term of the series
• \textbf{r}: the common ratio

This formula works only for infinite geometric series where the common ratio, \( r \), satisfies \( |r| < 1 \). If the common ratio is outside this range, the series does not converge and thus does not have a finite sum.

common ratio

The common ratio (\textbf{r}) in a geometric series is the constant value that each term is multiplied by to get the next term. It's calculated as:
\( r = \frac{a_2}{a_1} \)
where \( a_2 \) is the second term and \( a_1 \) is the first term. In our example,
\( a_2 = 2.8 \) and \( a_1 = 7 \),
thus \( r = \frac{2.8}{7} = 0.4 \). The common ratio tells us how quickly the terms are getting smaller or larger.

first term calculation

Identifying the first term (\textbf{a}) in a geometric series sets the foundation for calculating the sum of the series. To find the first term, substitute the first index value into the series expression. For our given series:
\( \text{The series is} \ \ \sum_{i=1}^{otegthinspaceotegthiotegthinspaceotegthinspaceotegthinspace\frac{otegthinspaceoteg}7}{ otegthinspaceototegthinspace (0.4)^{i-1} \;&n \}-1} \ \ (0.4)^{i-1} = ...of| \ = ...of| \ $ \ a_{s} = 7(1} = (0.4)^0 = ...1 = \ \ ing the first term = \The series' term \ a \ = \ \). :::::::::::::}}::::}} Prplayers ing the indice = }\0. \ \term.Originally found the finds:

infinite series solution

Solving an infinite geometric series involves substituting the known values for the first term (\textbf{a}) and the common ratio (\textbf{r}) into the sum formula:
\( S = \frac{a}{1-r} \)
For our specific series:

• \textbf{a} = 7
• \textbf{r} = 0.4

Substitute these values into the formula:
\( S = \frac{7}{1 - 0.4} = \frac{7}{0.6} \)
Simplify the expression to find:
\( S = \frac{7}{0.6} = \frac{7 \times 10}{0.6 \times 10} = \frac{70}{6} = \frac{35}{3} \)
Therefore, the sum of the infinite series is \( \frac{35}{3} \).