Hey there! Today I want to dive deep into the topic of finding the sum of a series. I know it can seem daunting at first, but trust me, once you understand the underlying principles, it becomes much easier. So let’s get started!
What is a series?
Before we jump into finding the sum of a series, let’s make sure we’re on the same page about what a series actually is. In mathematics, a series is defined as the sum of the terms of a sequence. In simpler terms, it’s a sequence of numbers that are added together.
Arithmetic Series
One type of series that you might come across is an arithmetic series. In an arithmetic series, each term is obtained by adding a constant difference to the previous term. For example, the series 2, 5, 8, 11, 14 is an arithmetic series with a common difference of 3.
Formula for finding the sum of an arithmetic series
Now, let’s talk about how to find the sum of an arithmetic series. There is a formula that can simplify this process:
Sum = (n/2) * (first term + last term)
In this formula, n represents the number of terms in the series. The first term is the initial term of the series, and the last term is the final term of the series.
Let’s take an example to see how this formula works in practice:
Find the sum of the arithmetic series: 1, 4, 7, 10, 13, … up to 25 terms.
Here, the first term is 1, the last term is 1 + (25-1) * 3 = 73, and the number of terms, n, is 25.
Plugging these values into the formula:
Sum = (25/2) * (1 + 73)
Simplifying further:
Sum = (25/2) * 74
Sum = 925
Geometric Series
Another type of series that you might encounter is a geometric series. In a geometric series, each term is obtained by multiplying the previous term by a constant ratio. For example, the series 3, 6, 12, 24, 48 is a geometric series with a common ratio of 2.
Formula for finding the sum of a geometric series
Similar to the arithmetic series, the geometric series also has a formula to find its sum:
Sum = (first term * (1 - common ratio ^ n)) / (1 - common ratio)
In this formula, n represents the number of terms in the series. The first term is the initial term of the series, and the common ratio is the constant ratio between consecutive terms.
Let’s take an example to illustrate this:
Find the sum of the geometric series: 3, 6, 12, 24, 48, … up to 5 terms.
Here, the first term is 3, the common ratio is 2, and the number of terms, n, is 5.
Plugging these values into the formula:
Sum = (3 * (1 - 2^5)) / (1 - 2)
Simplifying further:
Sum = (3 * (1 - 32)) / (1 - 2)
Sum = (3 * (-31)) / (-1)
Sum = 93
Conclusion
And there you have it! The sum of a series can be found using different formulas, depending on whether it’s an arithmetic series or a geometric series. Remember to carefully identify the type of series you’re dealing with and then apply the appropriate formula. Happy calculating!