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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 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 = 1,2,3,4,5 one 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.

Additional Information:
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 is 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 nn by n+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. .

Note:
1.Don’t forget to cross-check your answer.
2.The difference between any two consecutive terms in an A.P. is always the same and if it is not the same, then the given series is not an A.P.
3.(n−1)(n - 1)is the position of term in the sequence.