Question

Look at the sequence 5, 8, 13, 20…decide which of the following is the correct expression for the \[{n^{th}}\] term of the sequence
A.\[4n + 1\]
B.\[3n + 2\]
C.\[{n^2} + 4\]

Answer 1

Hint: Now the sequence given to us has 4 terms. Now to decide the formula for the \[{n^{th}}\] term we will check the options one by one. Putting 1, 2, 3 in place of n and checking whether we get the given sequence or not. The option that will give the sequence as the answer will be the correct option. So let’s check!

Complete step by step solution:
Option A: \[4n + 1\]
For n=1, \[4\left( 1 \right) + 1 = 4 + 1 = 5\] thus the first term is 5.
For n=2, \[4\left( 2 \right) + 1 = 8 + 1 = 9\] thus the second term is 9.
But in our sequence the second term is 8. Thus this is not the correct option.

Option B: \[3n + 2\]
For n=1, \[3\left( 1 \right) + 2 = 3 + 2 = 5\] thus the first term is 5.
For n=2, \[3\left( 2 \right) + 2 = 6 + 2 = 8\] thus the second term is 8.
For n=3, \[3\left( 3 \right) + 2 = 9 + 2 = 11\] thus the third term is 11.
But in our sequence the third term is 13. Thus this is not the correct option.

Option C: \[{n^2} + 4\]
For n=1, \[{\left( 1 \right)^2} + 4 = 1 + 4 = 5\] thus the first term is 5.
For n=2, \[{\left( 2 \right)^2} + 4 = 4 + 4 = 8\] thus the second term is 8.
For n=3, \[{\left( 3 \right)^2} + 4 = 9 + 4 = 13\] thus the third term is 13.
For n=4, \[{\left( 4 \right)^2} + 4 = 16 + 4 = 20\] thus the third term is 20.
Thus the given sequence can be obtained from this formula. So this is the correct option.
Option C is the correct option.
So, the correct answer is “Option C”.

Note: Note that we can solve this question in less time also with the method mentioned below. We can observe that the difference between the two terms of the given sequence is as 3,5,7…thus the difference between the first term and a term previous to it should be 1. Because the difference is the sequence of odd numbers.
\[
  8 - 5 = 3 \\
  13 - 8 = 5 \\
  20 - 13 = 7 \;
\]
So the term before 5 will be 4 with a difference of 1. But for that formula that will be the zeroth term. So now just check which formula answers 4 for n=0. That is only the third formula.
Hope this won’t complicate you!