Question

How do you find the explicit formula for the following sequence 5, 8, 11, 14, . . .?

Answer 1

Hint: As we can see that the given sequence is an arithmetic sequence i.e., the terms obtained is by adding or subtracting a constant to the preceding term, there is a constant difference between consecutive terms and hence by this, we can find the formula for the sequence.

Formula used:
\[{a_n} = {a_1} + \left( {n - 1} \right)d\]
\[{a_n}\]is the nth term
\[{a_1}\] is the first term
\[n\] is the terms number in the sequence
\[d\] is the common difference

Complete step by step solution:
This is an arithmetic sequence since there is a common difference between each term. In this case, adding 3 to the previous term in the sequence gives the next term.
An explicit formula of an arithmetic series includes all given information as
\[\Rightarrow {a_n} = {a_1} + \left( {n - 1} \right)d\]
This is the formula of an arithmetic sequence.
In the given arithmetic sequence, we get d as
\[\Rightarrow d = {a_2} - {a_1}\]
\[\Rightarrow d = 8 - 5 = 3\]
Substitute in the values of a1=5 and d=3 in
\[\Rightarrow {a_n} = {a_1} + \left( {n - 1} \right)d\]
\[\Rightarrow {a_n} = 5 + \left( {n - 1} \right)3\]
To simplify each term, apply distributive property
\[\Rightarrow {a_n} = 5 + 3n + 3\left( { - 1} \right)\]
Multiply 3 by -1 we get
\[\Rightarrow {a_n} = 5 + 3n - 3\]
Therefore, we get
\[\Rightarrow {a_n} = 2 + 3n\]
Hence, by applying this formula we can get the following sequence.

Note: An arithmetic progression (AP) is a sequence of numbers in which each succeeding number is obtained either by adding or subtracting a specific number called common difference. The general form of AP is: a, a + d, a + 2d, ….

 In an arithmetic sequence, the terms can be obtained by adding or subtracting a constant to the preceding term, there is a constant difference between consecutive terms, the sequence is said to be an arithmetic sequence, wherein in case of geometric progression each term is obtained by multiplying or dividing a constant to the preceding term.