Have you ever come across a set of numbers and been curious about how to calculate their sum? No need to worry, as I will walk you through a step-by-step guide on finding the sum of a series in this article.
Understanding Series
Before diving into finding the sum of a series, let’s first understand what a series is. In mathematics, a series is the sum of the terms of a sequence. A sequence is an ordered list of numbers that follow a specific pattern. Each term in the sequence is denoted by “a”, followed by a subscript. For example, a1, a2, a3, and so on.
Now, let’s move on to finding the sum of a series.
Summing an Arithmetic Series
If you have an arithmetic series, which means that each term is obtained by adding a constant difference to the previous term, you can easily find the sum using a formula:
Sn = (n/2)(a1 + an)
Where Sn is the sum of the first “n” terms, a1 is the first term, and an is the last term of the series.
Let’s take an example to clarify. Consider the series: 1, 4, 7, 10, 13. To find the sum of this arithmetic series, we need to determine the values of “n”, a1, and an.
In this case, the first term (a1) is 1, and the last term (an) is 13. The common difference between consecutive terms is 3 (4 – 1 = 3).
Using the formula, we can calculate the sum:
Sn = (5/2)(1 + 13) = 35
Therefore, the sum of the series 1, 4, 7, 10, 13 is 35.
Summing a Geometric Series
Now, let’s move on to finding the sum of a geometric series. In a geometric series, each term is obtained by multiplying the previous term by a constant ratio. The formula for finding the sum of a geometric series is:
Sn = a1 * (1 - rn) / (1 - r)
Where Sn is the sum of the first “n” terms, a1 is the first term, and r is the common ratio between consecutive terms.
Let’s take an example. Consider the series: 3, 6, 12, 24, 48. To find the sum of this geometric series, we need to determine the values of “n”, a1, and r.
In this case, the first term (a1) is 3, and the common ratio (r) is 2 (6/3 = 2). Since we don’t have the last term, we need to find “n”, which represents the number of terms in the series.
We can find “n” by using the formula:
an = a1 * rn-1
Let’s find “n” by substituting the values we know:
48 = 3 * 2n-1
By simplifying the equation, we find that “n” is equal to 5.
Now, we can use the formula for the sum of a geometric series:
Sn = 3 * (1 - 25) / (1 - 2) = 93
Therefore, the sum of the series 3, 6, 12, 24, 48 is 93.
Conclusion
Calculating the sum of a series can be quite straightforward once you understand the underlying patterns and formulas. In this article, we explored how to find the sum of both arithmetic and geometric series. Remember to identify the type of series you’re dealing with and apply the appropriate formula. Happy calculating!