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Introduction to Sequences and Series
Sequence and series are two of the most fundamental ideas in arithmetic. A sequence is an itemized list of components that allows for an unlimited number of repeats, whereas a series is the sum of all items. An arithmetic progression is one of the most common examples of sequence and series.
Define the concept of sequence.
A sequence is a collection of objects/events organized sequentially so that each item occurs before or after every other member.
Imagine that you are in a pizza shop. You have made a purchase, and your order number is 282. The shop is currently serving order number 275. So, how many orders do you believe they will cater to before your number?
The correct answer is six. You have to wait for six more orders before your turn because you are in a sequence.
Define the concept of series.
While a sequence is a list of items/objects that have been sorted in a specific order, a series can be broadly defined as the sum of all the terms in an addition or multiplication order, as well as any other mathematical operations. All the sequence’s terms, however, must have a clear link.
You can understand the principles better by solving questions based on the formulas. They are quite similar to sets, but the main difference is that sequence terms can repeatedly recur in different positions.
Sequences: Finite and Infinite
A sequence is a collection of integers arranged in a specific order. It is a series of integers that follow a precise pattern, and the terms refer to all the elements that make it up. Consider the following order of events:
[1, 3, 5, 7, 9, 11, and so on….]
Because we know it consists of odd natural integers, we may call it a sequence. In this situation, the number of terms in the series will be infinite. A series with an unlimited number of terms is known as an infinite sequence. But what if you decided to put an end to it?
(1), (2), (3), (5), (7), (8),…(131)
In the above series, you can count the number of terms if 131 is the last term. As a result, finite sequences have a fixed number of countable terms. As previously stated, terms such as 1, 3, 5, 7, and so on are these terms.
Types of Sequences
An arithmetic sequence is when the difference between any two successive terms in an arithmetic (linear) sequence is constant.
A quadratic sequence is a numerical sequence where the second difference between any two consecutive terms is constant.
A geometric sequence is a number series where you can find any term following the first by multiplying the previous term by a constant, non-zero value known as the standard ratio.
Fibonacci Sequence
The Fibonacci sequence is unique in that the first two terms are fixed. When we discuss terms, we can see a broad depiction of these terms in sequences. A term is denoted by an, where n is the nth term.
The first term is defined as a1 = 1 and a2 = 1. Every term in this Fibonacci sequence equals the sum of the preceding two terms beginning with the third term.
As a result, a3 will be supplied as a1 + a2
a3 = 1+1 = 2.
Likewise, a4 = a2 + a3 = 1 + 2 = 3
a5 = a3 + a4 = 2 + 3 = 5
Therefore if we want to write the Fibonacci sequence, we will write it as, [1, 1, 2, 3, 5,…].
In general, we can say,
an = an-1 + an-2, where the value of n ≥ 3.
Series
A series is the sum of all sequences. As the sequence is either finite or infinite, the series is either finite or infinite. Sigma is the symbol used for series, which indicates that summation is involved. A series S, for example, can be,
Sum (1, 3, 5, 7, 9, 11,…)
Formulae
A list of some basic formulae of arithmetic progression and geometric progression:
| Arithmetic Progression | Geometric Progression |
Sequence | a, a+d, a+2d,……, a+(n-1)d,…. | a, ar, ar2,….,ar(n-1),… |
Common Difference or Ratio | Successive term – Preceding term Common difference = d = a2 – a1 | Successive term / Preceding term Common ratio = r = ar(n-1) / ar(n-2) |
General Term (nthterm) | an = a + (n-1)d | an = ar(n-1) |
nth term from the last term | an = l – (n-1)d | an = 1 / r(n-1) |
Sum of first n terms | sn = n/2(2a + (n-1)d) | sn = a(1 – rn) / (1 – r) if r < 1 sn = a(rn -1) / (r – 1) if r > 1 |
Examples
- Given a sequence of 4,7,10,13,16,19,22……, find: a) a common difference; b) The twenty-first term
The supplied sequence is 4, 7, 10, 13, 16, 19, 22, etc.
7 – 4 = 3 is the average difference.
Tn denotes the nth term of the arithmetic sequence, which may be calculated as:
Tn = a + (n-1)d
Where “a” is the first term, and “d” is a common difference.
Therefore,
Tn = 4 + (n – 1)3
Tn = 4 + (n – 1)3
Tn = 4 + 3n – 3
Tn = 3n + 1
Now
T21 = 3*21+1
T21 = 63+1 = 64
2: If an A.P.’s 9th term is zero, find the ratio of its 29th and 19th terms.
Solution: d = 0 9th term = a + (9 – 1) => 8d + a = 0
Alternatively, a = -8d
Now,
(a + 28d) / (a + 18d) = ((-8d + 28d)) / ((-8d + 18d)) = 20d / 10d = 2/1 (29th term)/(19th term)
The ratio is 2:1.
Conclusion
A sequence is defined as a set of integers that are created according to a set of criteria. This article provides a clear introduction to sequence and series.
A sequence is a collection of numbers in a specific order that follows a pattern. The number of terms in a sequence or series can be finite or infinite, depending on the number of terms in the sequence or series. The terms of a sequence can be added or subtracted to form a series. The nth term, represented by an or Tn, is a sequence/series general term.
