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One of the primary concepts of sums of ap is sequence and series. A sequence is an itemised collection of elements that allows for any number of repetitions, whereas a series is the total number of elements. A common example of sequence and series is an arithmetic progression. By solving examples using the formulas, one can better understand the principle. They’re similar to sets, but the main difference is that individual terms might appear multiple times in different positions in a sequence. The number of terms determines the length of a series, which might be finite or infinite. In Class 11 Maths, this topic is explained thoroughly. The ideas will be discussed using definitions, formulas, and questions on sequence and series.
Definition of sequences and series
A sequence is an arrangement of things or a set of numbers arranged in a specific order and according to a set of rules. If the terms of a sequence are a1, a2, a3, a4,……, etc., then the position of the term is 1,2,3,4,….. The number of terms defines a finite or infinite sequence.
If a1, a2, a3, a4,…… is a sequence, the corresponding series is
SN = a1+a2+a3 + .. + aN
Note: The series is defined depending on whether or not the sequence is finite or infinite.
Types of sequence and series
Sequences can be found in a variety of forms mainly:
1. Geometric sequences
2. Arithmetic sequences
3. Harmonic sequences
4. Fibonacci numbers
Geometric Sequences
A geometric sequence is a series of numbers in which each term is formed by multiplying or dividing a defined integer by the last number.
Arithmetic sequences
An arithmetic sequence is one in which each term is formed by adding or subtracting a specific number from the one before it.
Harmonic sequences
When the reciprocals of all the elements of a number sequence form an arithmetic sequence, it is said to be in a harmonic sequence.
Fibonacci numbers
Fibonacci numbers are a fascinating number series in which each element is formed by adding two preceding elements, and the sequence begins with 0 and 1. F0 = 0 and F1 = 1; Fn = Fn-1 + Fn-2.
Formulas for sequence and series
Some basic arithmetic and geometric progression formulas for sequence and series are listed below.
Arithmetic Progression | Geometric Progression | |
Sequence | an 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 (nth Term) | 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 = an (rn -1)/(r – 1) if r > 1 |
Difference between sequence and series
Sequence | Series |
A sequence is a logical arrangement of objects, with each member appearing before or after the others. | A series is the sum of ap, a group of terms. A series, in other terms, is a group of numbers linked together by addition operations. |
The most significant factor is the order in which the elements appear. | In a series, the order of the elements is irrelevant. |
A sequence’s elements follow a defined pattern. | The series is the sum of ap in the items in the sequence. |
It’s crucial to pay attention to the sequence’s order. As a result, a series of (3, 4, 5) varies from a sequence of (3, 4, 5). (5, 4, 3\). | In the series, (3+4+5) is the same as (5+4+3). The sequence’s order isn’t important. |
Example: \(1, 2, 3, 4,…\) is a sequence | Example: \(1+2+3+4,…\) is a series |
Examples/Questions on sequence and series.
1.: If 4,7,10,13,16,19,22……is a sequence, Find:
1. Common difference
2. nth term
3. 21st term
Solution: Given sequence is, 4,7,10,13,16,19,22……
a) The common difference = 7 – 4 = 3
b) Tn = a + (n-1)d, where “a” is the first term and d is a common difference, is the nth term of the arithmetic series.
c) 21st term as T21 = 4 + (21-1)3 = 4+60 = 64.
2. Consider the numbers 1, 4, 16, 64, 256, 1024, and so on….. Determine the 9th term and the common ratio.
Solution:
The common ratio (r) = 4/1 = 4
To get the following term, multiply the previous term by four.
Tn = ar denotes the nth term of the geometric sequence (n-1)
R is the common ratio, while an is the first word.
Here a = 1, r = 4, and n = 9
Conclusion
A sequence is a logical grouping of elements (or events) that occur after the other. The sum of ap, a set of words, is defined as a series. In other words, a series is a collection of numbers linked together by addition operations. The definitions, kinds, formulas, differences, and sequence and series applications are discussed above. However, above we have mentioned all the detailed information related to the sum of ap, formulas for sequence and articles. Moreover, the basic questions on sequence and series are also mentioned for easy understanding.
So, the 9th term is can be calculated as T9 = 1* (4)(9-1)= 48 = 65536.
