Sequences and series might look similar, but they are not the same. Let's look at the differences.
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Jetzt kostenlos anmeldenSequences and series might look similar, but they are not the same. Let's look at the differences.
A sequence can be described as a set of numbers, known as terms, that all follow a rule. They are listed in a specific order, and the rule they follow is usually a mathematical pattern. Here are some examples of sequences and the rule explained;
\((3, 9, 15, 21, 27, 33) \) Increasing by 6
\((72, 64, 56, 48, 40, 32)\) Decreasing by 8
\((5, 10, 20, 40, 80, 160)\) - Multiplying by 2
Sequences can be either finite like the examples above or infinite, meaning they have no end; they can be shown like this;
\((1, 2, 3, 4, 5, 6, \dots) \)
\((4, 7, 10, 13, 16, \dots)\)
Due to these sequences being infinite, we can use a formula to find a specific term rather than going through the whole sequence. We will look at some of these formulas later in this article.
There are two different types of common sequences;
Arithmetic sequences - in this sequence, the terms increase or decrease by addition or subtraction. This difference is constant and known as the common difference or \(d\) .
Geometric sequences - in this type of sequence, the terms increase or decrease by multiplication or division. This difference is known as the common ratio or \(r\) .
A series is an addition of the terms within a sequence, for example;
\((3, 9, 15, 21, 27, 33)\) is a sequence and its series is \(3 + 9 + 15 + 21 + 27 + 33\)
\((72, 64, 56, 48, 40, 32)\) is a sequence and its series is \(72 + 64 + 56 + 48 + 40 + 32\)
When working with sequences and series, you may be asked to find a specific term within a sequence or the sum of a series. Here are the formulas that you can use to help you find the answers:
There is a formula for both types of sequences, arithmetic and geometric. The formula used for finding the \(n\)th term in an arithmetic sequence is;
\[ u_n = a + (n-1)d\]
Let's have a look at an example and how we would substitute it into the formula;
Find the fifteenth term of this sequence \((5, 12, 19, 26, 33, 40, \dots )\)
The formula used for finding the \(n\)th term in a geometric sequence is;
\[ u_n = ar^{n-1}\]
The common ratio is the number used to multiply or divide each term.
Let's have a look at an example and how we would substitute it into the formula;
Find the \(24\)th term of this sequence \( (6, 12, 24, 48, 96, \dots ) \)
The formula used for finding the sum of the first terms of an arithmetic series is;
\[ s_n = \frac{n}{2}(2a + (n-1)d) \]
Let's have a look at an example and how we would substitute it into the formula;
Find the sum of the first \(35\) terms of this series \( (2, 8, 14, 20, 26, 32, \dots ) \)
Two different formulas can be used when finding the sum of a geometric series. The first one is easier to use when \(r <1\) and the second when \(r> 1\);
\[ s_n = \frac{a(1-r^n)}{1-r}\]
or
\[ s_n = \frac{a(r^n-1)}{r-1} \]
Let's have a look at an example and how we would substitute it into the formula;
Find the sum of the first \(50\) terms of this series \( (4, 12, 36, 108, \dots ) \)
The Greek letter sigma can be used to identify the sum. To use this, write the limits above and below sigma to show the terms you are using. This is shown below;
\[ \sum\limits_{r=1}^6 (2r+4) \]
This shows you that you will be finding the sequence by substituting r into the equation from \(1\) to \(6\). This will then give you the figure to create your sum.
Sequences and series can be applied in many real-life situations, and this is also known as modeling. Many examples come from money - for example if someone saves £ 10 in first month and in second month the person saves double what he had saved in previous month and so on. This way we can use geometric series to find his savings in an year.
Dave deposits £ 10 on his bank account in the first month, in the second month he deposits £ 20 which is the double what he had deposited in the previous month and he continuous to do so for an year. How much money will he deposit in an year?
Here as per the data, Dave deposits £ 10 in first month, £ 20 in second month, £ 40 in third month and so on.
Therefore the series becomes \(10, 20, 40, 80, \dots \), up to 12 terms (1 year = 12 months)
Here
\[ \begin{align} & a = 10 \\ & r =2 \\ & n = 12 \end{align}\]
Since \(r > 1\) we can use the formula,
\[ \begin{align} s_n &= \frac{a (r^n-1)}{r-1} \\ &= \frac{10(2^{12}-1)}{2-1} \\ &= \frac{10(4096-1)}{1} \\ &= 40950\end{align}\]
This means Dave will have deposited £ 40950 in his account in an year .
A sequence is a set of numbers that follow a specific rule
There are two different types of sequences, arithmetic and geometric
A series is the sum of a sequence
Sequences and series can be modeled into real-life scenarios.
A sequence is a set of numbers that follow a mathematical rule. A series is the sum of a sequence.
The difference between a sequence and a series is that a sequence is the set of numbers, whereas the series is the addition of those numbers.
There are formulae you can use to solve each type of sequence and series.
What is a sequence?
A sequence is a set of numbers that follow a specific rule
What is a series?
A series is the sum of a sequence
What are the two types of sequences?
Arithmetic and geometric
When do you use the sigma notation?
This notation is used to identify the sum of the series.
What is the binomial expansion used for?
Expanding out things like \((x+y)^n\) without having to do all the multiplication.
What is the simplified form of the binomial expansion formula using summation notation?
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