How do you calculate series?

Calculating a series depends largely on the type of series you are dealing with. A common type of series to calculate is an arithmetic series, which will be explained in detail below.

Understanding Arithmetic Series

An arithmetic series is the sum of the terms in an arithmetic sequence. An arithmetic sequence is a list of numbers where the difference between any two consecutive terms is constant. This constant difference is known as the 'common difference', denoted by 'd'. The first term of the sequence is denoted by 'a'.

Key Formulas

The reference text provides the key formulas needed for calculating arithmetic sequences and series:

nth term of an arithmetic sequence (an): an = a + (n - 1)d
- Where 'a' is the first term, 'd' is the common difference, and 'n' is the term number.
Sum of an arithmetic series (Sn):
- Sn = n/2 (2a + (n - 1) d)
- OR
- Sn = n/2 (a + an)
  - Where 'n' is the number of terms, 'a' is the first term, and 'an' is the nth term.

How to Calculate the Sum of an Arithmetic Series

Here's a step-by-step process for calculating the sum of an arithmetic series:

Identify 'a', 'd', and 'n':
- Determine the first term ('a') of the sequence.
- Determine the common difference ('d') between consecutive terms.
- Determine the number of terms ('n') you want to sum.
Choose the appropriate formula:
- If you know the first term ('a'), the common difference ('d'), and the number of terms ('n'), use the formula: Sn = n/2 (2a + (n - 1) d).
- If you know the first term ('a'), the last term ('an'), and the number of terms ('n'), use the formula: Sn = n/2 (a + an).
Plug in the values:
- Substitute the values of 'a', 'd', 'n', or 'an' into the chosen formula.
Calculate the sum:
- Solve the equation to find the sum of the series (Sn).

Example

Let's calculate the sum of the first 10 terms of the arithmetic series where the first term (a) is 2 and the common difference (d) is 3.

Given:
- a = 2
- d = 3
- n = 10
Using the formula: Sn = n/2 (2a + (n - 1) d)
Substitute the values: S10 = 10/2 (2*2 + (10 - 1) * 3)
Calculate: S10 = 5 (4 + 9 * 3) = 5 (4 + 27) = 5 * 31 = 155
- The sum of the first 10 terms is 155.

Other Types of Series

While the formulas provided are specific to arithmetic series, there are other types of series, such as geometric series, that have different calculation methods.

Geometric Series

A geometric series is a sum of terms in a geometric sequence, where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the 'common ratio'. The sum of a geometric series is calculated using different formulas than arithmetic series.

Formula

*   `Sn = a (1 - r^n) / (1-r)`  where *r* is common ratio

Summary

Calculating a series involves applying specific formulas based on the nature of the series, be it arithmetic, geometric, or other. For arithmetic series, use the provided formulas to find the sum based on the first term, common difference, and number of terms.