Sequences
Listing out the collection of elements (numbers or alphabets) horizontally separated by commas form a sequence. Usually, the collection of elements is ordered in such a way that each element is identified first member, second member, third member and so on. For example, list of monthly profits of a business for the past 1 year. The amount of money is being deposited deposited in a bank account every year over a number of years form a sequence. We have applications of sequences in many situations of our day to day life.If a sequence follows a specific pattern, it is called progression. For example, arithmetic and geometric progrssesions are sequences which follow specific patterns.
Problem 1 :Write the first four terms of the sequence whose nth term is given below.an = 2n + 5Solution :To get the first four terms, substitute n = 1, 2, 3, and 4 into the nth term of the given sequence.n = 1 :a1 = 2(1) + 5= 2 + 5= 7n = 2 :a2 = 2(2) + 5= 4 + 5= 9n = 3 :a3 = 2(3) + 5= 6 + 5= 11n = 4 :a4 = 2(4) + 5= 8 + 5= 13The first four terms of the given sequence :7, 9, 11, 13Problem 2 :Write the first three terms of the sequence whose nth term is given below.
Solution :n = 1 :
n = 2 :
n = 3 :
The first three terms of the given sequence :
Problem 3 :Find the 25th term of the sequence whose nth term is given below.an = (n – 1)(2 – n)(n + 3)Solution :To find the 25th term of the given sequence, substitute 25 for n into the nth term of the sequence.a25 = (25 – 1)(2 – 25)(25 + 3)a25 = (24)(-23)(28)a25 = -15456The 25th term of the given sequence is -15456.Problem 4 :Let the sequence an be defined as follows.a1 = 1, an = an – 1 + 2 for n ≥ 2Find first five terms.Solution :The first term of the sequence is given : a1 = 1.n = 2 :a2 = a2 – 1 + 2= a1 + 2= 1 + 2= 3n = 3 :a3 = a3 – 1 + 2= a2 + 2= 3 + 2= 5n = 4 :a4 = a4 – 1 + 2= a3 + 2= 5 + 2= 7n = 5 :a5 = a5 – 1 + 2= a4 + 2= 7 + 2= 9The first five terms of the given sequence :1, 3, 5, 7, 9
Series
If you add all the terms of a sequence, it is called series.Let a1, a2, a3, …….., an – 1, an, be a given sequence, then the expressiona1 + a2 + a3 + …….. + an – 1 + anis called the series associated with the given sequence.The series is finite or infinite according as the given sequence is finite or infinite. Series are often represented in compact form, called sigma notation, using the Greek letter ∑ (sigma) as means of indicating the summation involved. Thus, the series a1 + a2 + a3 + …….. + an – 1 + anis abbreviated as
where i = 1, 2, 3, ……., n.We have different formulas to evaluate a series depending on its type. To know more about the formulas to evaluate a series and sample problems, please click on the following links.1. Arithmetic series2. Finite geometric series3. Infinite geometric series4. Sum of first n natural numbers5. Sum of squares of first n natural numbers6. Sum of cubes first n natural numbers7. Sum of first n even natural numbers8. Sum of first n odd natural numbers
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