A series is the sum of the terms in a sequence. In mathematics, there are two main types of series: arithmetic and geometric. Each type has a set of formulas that can be used to find the sum of specific terms within the series.
An arithmetic series is the sum of an arithmetic sequence, where each term in the sequence increases or decreases by a constant value. For instance, the sequence 3, 7, 11, 15, 19 has a constant difference of 4, and its arithmetic series would be 3 + 7 + 11 + 15 + 19. To quickly find the sum of an arithmetic series, you can use the formula:
S = (n/2)(2a + (n-1)d)
Here, S represents the sum, n is the number of terms, a is the first term, and d is the common difference.
To understand the proof of this formula, follow these steps:
First, write out the terms of the sum:
3 + 7 + 11 + 15 + 19
Next, rewrite the sum in reverse order:
19 + 15 + 11 + 7 + 3
Add the two sums together:
(3 + 19) + (7 + 15) + (11 + 11) + (15 + 7) + (19 + 3) = 22 + 22 + 22 + 22 + 22 = 110
To find the average of the sums, divide by 2:
110/2 = 55
This is equivalent to our formula for the sum of an arithmetic series, making it easier to find the sum without manually adding each term. For example, let's find the sum of the first 30 terms of the arithmetic series: 5 + 8 + 11 + 14 + 17 ...
To solve this, identify the variables and plug them into the formula:
S = (30/2)(2*5 + (30-1)*3) = 15(10 + 87) = 15(97) = 1455
The sum of the first 30 terms in this arithmetic series is 1455.
A geometric series is the sum of a geometric sequence, where each term is multiplied or divided by a common ratio. For example, the sequence 2, 4, 8, 16, 32 has a common ratio of 2, and its geometric series would be 2 + 4 + 8 + 16 + 32. There are two formulas that can be used to find the sum of a geometric series, depending on the value of the common ratio.
If the common ratio is greater than 1, use this formula:
S = (a(1-r^n))/(1-r)
If the common ratio is less than 1, use this formula:
S = (a(r^n-1))/(r-1)
To understand the proof of these formulas, follow these steps:
First, write out the terms of the sum:
2 + 4 + 8 + 16 + 32
Next, subtract the second sum from the first:
(2 + 4 + 8 + 16 + 32) - (2 + 4 + 8 + 16) = (2*32) + (4*32) + (8*32) + (16*32) - (2 + 4 + 8 + 16) = 2(1 + 2 + 4 + 8) = 31*2 = 62
Rearrange the sum by removing the common factor:
2((1-2^5)/(1-2)) = 2((1-32)/-1) = 2(31) = 62
Let's try this with an example. Find the sum of the first 15 terms of the series: 1 + 3 + 9 + 27 + 81 ...
To solve this, identify the variables and plug them into the formula for r > 1:
S = (1(1-3^15))/(1-3) = (1(-14348907))/-2 = 7174453.5
The sum of the first 15 terms in this geometric series is 7174453.5.
Sigma notation is the use of the Greek letter sigma (Σ) to represent the sum of a series. This is useful when there are a large number of terms in a series, making it challenging to write out the entire sum. The sigma symbol indicates that the variable of the expression within the brackets is to be substituted with values from the given range. For example:
Σ(r^k) from k=1 to k=7
This means that r will be substituted with values starting from 1 and ending at 7, giving us a total of 7 terms in the series. Once the terms have been identified, you can use the appropriate formula to find the sum of the series.
Series can also be applied to real-life situations, particularly those involving money, such as savings or salaries. If the amount increases by a fixed amount each time, it would create an arithmetic sequence, and its total savings would be the arithmetic series. On the other hand, if the amount increases by a percentage, it would create a geometric sequence, and its total savings would be the geometric series.
For instance, let's say John saves £150 in the first month and increases his savings by £20 each month. This would result in an arithmetic sequence, and if he wanted to know his total savings after 2 years, it would be the arithmetic series.
To determine the total number, follow these steps:
First, identify the appropriate formula to use based on the given series.
S = (n/2)(2a + (n-1)d)
Next, substitute the corresponding values from the question into the formula:
S = (24/2)(2*150 + (24-1)*20) = (12)(300 + 460) = 12(760) = 9120
Therefore, after 2 years, John will have a total of £9120 saved.
In math, an arithmetic series is a type of series formed by adding or subtracting a constant number to each term in a sequence. This creates a series where each term is the result of adding the previous term and the constant number. Understanding arithmetic series is crucial for solving complex mathematical problems and equations.
There are several formulas available for determining the sum of a series, depending on the type of series. For arithmetic series, the most commonly used formula is Sn = n/2(2a + (n-1)d), where n represents the total number of terms in the series, a is the first term, and d is the common difference between terms. This formula can be applied to any arithmetic series to calculate its sum.
On the other hand, for geometric series, there are two different formulas that can be used based on the common ratio. If the common ratio is less than 1, the formula is Sn = a(1-r^n)/1-r, where a is the first term and r is the common ratio. For a common ratio greater than 1, the formula is Sn = a(r^n-1)/r-1. It is crucial to use the appropriate formula to arrive at the correct sum of the series.
Many individuals often confuse a sequence and a series in mathematics. A sequence is a set of numbers that follow a particular rule or pattern and are arranged in a specific order. On the other hand, a series is the sum of terms within a sequence. Simply put, a sequence is a list of numbers, while a series is the outcome of adding those numbers together.
