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Question: How do you write the first five terms of the sequence \({a_n} = 3n + 1\)?...

How do you write the first five terms of the sequence an=3n+1{a_n} = 3n + 1?

Explanation

Solution

In order to determine the first five terms of the sequence put the value of nnfrom 1 to 51\,to\,5one by one to find the values of a1,a2,a3,a4,a5{a_1},{a_2},{a_3},{a_4},{a_5}.From the terms you can clearly see that that the given sequence is an Arithmetic Progression(A.P) having first term as a=4a = 4 and common difference as d=3d = 3.

Complete step by step solution:
We are given a sequence an=3n+1{a_n} = 3n + 1
The first five terms of the sequence are a1,a2,a3,a4,a5{a_1},{a_2},{a_3},{a_4},{a_5}.

In order to determine the first five terms of the above sequence we have to substitute the value of n=1,2,3,4,5n = 1,2,3,4,5one by one.

When n=1n = 1
a1=3(1)+1=3+1=4{a_1} = 3(1) + 1 = 3 + 1 = 4

When n=2n = 2
a2=3(2)+1=6+1=7{a_2} = 3(2) + 1 = 6 + 1 = 7

When n=3n = 3
a3=3(3)+1=9+1=10{a_3} = 3(3) + 1 = 9 + 1 = 10

When n=4n = 4
a4=3(4)+1=12+1=13{a_4} = 3(4) + 1 = 12 + 1 = 13

When n=5n = 5
a5=3(5)+1=15+1=16{a_5} = 3(5) + 1 = 15 + 1 = 16

By looking at the terms we can clearly say that the given sequence is an Arithmetic Progression (A.P.) Having first term a=4a = 4and common difference(d) as d=3d = 3.
Therefore, the first five terms of the sequence A.P. an=3n+1{a_n} = 3n + 1are 4,7,10,13,164,7,10,13,16.

Note: 1.Sequence: A sequence is a function whose domain is the set of N of natural numbers.
2.Real Sequence: A sequence whose range is a subset of R is called a real sequence.
In other words, a real sequence is a function having domain N and range equal to a subset of the set R of real numbers.
3.Arithmetic Progression (A.P): A sequence is called an arithmetic progression if the difference of a term and the previous term are always the same.

i.e. an+1−an={a_{n + 1}} - {a_n} = constant (=d)( = d)for all n∈Nn \in N.

The constant difference is generally denoted by d which is called as the common difference.
In order to determine whether a sequence is an A.P. or not when its nth term is given, we may use the following algorithm .

Algorithm:

Step 1: Obtain an{a_n}.

Step 2: Replace nnbyn+1n + 1in an{a_n}to get an+1{a_n} + 1

Step 3: Calculate an+1−an{a_{n + 1}} - {a_n}.

Step 4: If an+1−an{a_{n + 1}} - {a_n}is independent of n , the given sequence is an A.P.Otherwise is not an A.P. .