Question

Find the sum of the series. $$ 3+15+75+375+\cdots+3\left(5^{8}\right) $$

Short answer

Answer: The sum of the given geometric series is \(1464843\).

Step-by-step solution

Identify the first term and the common ratio

In this geometric series, it's clear that the first term (\(a_1\)) is 3. Observe that every term is obtained by multiplying the previous term by 5. So, the common ratio (\(r\)) is 5.

Find the number of terms in the series

The series goes up to \(3\left(5^{8}\right)\), and we know that the first term is \(3(5^0)\). To find the total number of terms (\(n\)), we must find which term \(3\left(5^{8}\right)\) represents. Since each term can be represented as \(3(5^k)\), where \(k\) starts at 0, we have: $$ 3\left(5^{8}\right) = 3(5^k) $$ In this case, \(k=8\). As terms start from 0, we have 9 terms in total (0 to 8). So, \(n=9\).

Use the geometric series formula to find the sum

Now that we have the first term (\(a_1=3\)), the common ratio (\(r=5\)), and the number of terms (\(n=9\)), we can use the formula for the sum of a geometric series: $$ S_n = \frac{a_1(1-r^n)}{1-r} $$ Plug in the values to get: $$ S_9 = \frac{3(1-5^9)}{1-5} $$ Calculate the result: $$ S_9 = \frac{3(1-1953125)}{1-5} = \frac{3(-1953124)}{-4} =\frac{5859372}{4}=1464843 $$ So, the sum of the given geometric series is \(1464843\).

Key concepts

Common Ratio

In a geometric series, the common ratio is a central concept. It is the number by which you multiply each term to get the next term in the series. For this exercise, if you start with the number 3 and multiply it by 5, you get 15. Then, if you multiply 15 again by 5, you get 75, and this pattern continues. This multiplication factor, which is consistent throughout the series, is known as the common ratio, denoted by \(r\).

• In the given series, the common ratio \(r\) is 5.
• This means that each term is 5 times the term before it.

Understanding the common ratio allows you to predict any term in the series by multiplying the previous terms enough times. It's a simple but powerful concept that makes solving problems like this much easier.

First Term

The first term in a geometric series plays an essential role as it acts as the starting point for the series. In every series, the first term is denoted by \(a_1\). In the exercise we have, the series begins with the number 3. This is our first term. It's important because:

• The entire series builds on this initial number.
• It serves as a baseline for calculating subsequent terms using the common ratio.

In formulas and calculations, the first term is crucial because it helps determine the entire series. When plugging into formulas, like the sum formula, knowing \(a_1\) is essential to perform accurate calculations. Without the first term, the pattern established by the common ratio doesn't even have a starting point!

Series Sum Formula

The series sum formula is a mathematical expression that simplifies finding the total of all terms in a geometric series. Calculating a large number of terms individually would take a lot of time. Instead, we use the formula:\[ S_n = \frac{a_1(1-r^n)}{1-r} \]This equation offers an efficient method to compute the sum, \(S_n\), of the first \(n\) terms in a geometric series:

• \(a_1\) is the first term.
• \(r\) is the common ratio.
• \(n\) represents the number of terms in the series.

In this exercise, with \(a_1 = 3\), \(r = 5\), and \(n = 9\), plugging these values gives the sum as \(S_9 = 1464843\). Knowing this formula allows you to focus on the series structure and make swift calculations instead of manual tedious additions.