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Question: How do you write the expression for the \({n^{th}}\) term of the sequence given? \(1, 4, 7, 10, 13...

How do you write the expression for the nth{n^{th}} term of the sequence given?
1,4,7,10,13,...1, 4, 7, 10, 13,...

Explanation

Solution

We will first identify the first term of the given sequence and the type of sequence. Then, we will find the difference between successive terms. Finally, we will use the formula to find the nth{n^{th}} term of the sequence and simplify it into an expression.

Formula used:
an=a+(n−1)d{a_n} = a + (n - 1)d, where an{a_n} is the nth{n^{th}} term of the sequence, aa is the first term, dd is the common difference, and nn is the number of terms.

Complete step by step solution:
The sequence given to us is 1,4,7,10,13,...1,4,7,10,13,....
We have to find an expression for the nth{n^{th}} term of the above sequence.
We have a=a1=1a = {a_1} = 1, which is the first term of the sequence.
Next, we have a2=4{a_2} = 4, which is the second term of the sequence.
Let us check the difference between the first and the second term. We have
a2−a1=4−1=3{a_2} - {a_1} = 4 - 1 = 3
Now, the third term is a3=7{a_3} = 7. The difference between the second and the third term is
a3−a2=7−4=3{a_3} - {a_2} = 7 - 4 = 3
This difference is the same as that of the first and the second term.
Hence, the sequence given to us is an arithmetic progression. The common difference between the successive terms is d=3d = 3.
To find the nth{n^{th}} term, we will use the general form of the nth{n^{th}} term of an arithmetic progression,
an=a+(n−1)d{a_n} = a + (n - 1)d.
Substituting a=1a = 1 and d=3d = 3 in the formula, we have
an=1+(n−1)3{a_n} = 1 + (n - 1)3
Multiplying the terms, we get
⇒an=1+3n−3\Rightarrow {a_n} = 1 + 3n - 3
Subtracting the like terms, we get
⇒an=3n−2\Rightarrow {a_n} = 3n - 2
Let us test this expression.
The fourth term must be
a4=3(4)−2=10{a_4} = 3(4) - 2 = 10

So, we can say that the expression is correct, as the fourth term in the given sequence is 10.

Note:
An arithmetic progression is a sequence of numbers such that the difference of any two successive members is a constant. In the given sequence, the common difference between two successive terms is 3. If the successive terms have a common ratio between them then the progression is known as geometric progression. The reciprocal of an arithmetic series or progression is known as harmonic progression.