Question

How do you find the expression for the $ {n^{th}} $ term in the sequence $ 3,8,15,24 \ldots $ ?

Answer 1

Hint: In order to determine the $ {n^{th}} $ term rule for the sequence $ 3,8,15,24 \ldots $ , we will determine the sequence of the consecutive term, then we will determine the suitable form of the $ {n^{th}} $ term of the given sequence and also verify it.

Complete step-by-step answer:
Consider the sequence $ 3,8,15,24 \ldots $ .
The differences between consecutive pairs of terms are the consecutive odd numbers.
 $ 0 + 3 = 3 $
 $ 3 + 5 = 8 $
 $ 8 + 7 = 15 $
 $ 15 + 9 = 24 $
Hence, the sequence can be rewritten in the form, $ 0 + 3,3 + 5,8 + 7,15 + 9, \ldots $ .
Thus, we can say that the differences between consecutive pair of terms are the consecutive odd numbers i.e., $ 3,5,7,9 \ldots $
Then, the next upcoming terms will be $ 24 + 11 = 35 $ , $ 35 + 13 = 48 $ , so on.
We need to determine the $ {n^{th}} $ term of the sequence.
Thus, now we will find the suitable expression for the $ {n^{th}} $ term by the suitable term,
 $ {a_n} = {\left( {n + 1} \right)^2} - 1 $
 $ {a_n} = {n^2} + 2n + 1 - 1 $
 $ {a_n} = {n^2} + 2n $
Now, let us verify this term, by substituting the value of $ n $ from $ 1 $ ,
 $ {a_1} = {1^2} + 2\left( 1 \right) = 3 $
 $ {a_2} = {2^2} + 2\left( 2 \right) = 8 $
 $ {a_3} = {3^2} + 2\left( 3 \right) = 15 $
 $ {a_4} = {4^2} + 2\left( 4 \right) = 24 $
And so on…
Hence, the expression for the $ {n^{th}} $ term in the sequence $ 3,8,15,24 \ldots $ is $ {a_n} = {n^2} + 2n $ .
So, the correct answer is “${a_n} = {n^2} + 2n $”.

Note: A sequence can be defined as a function whose domain is either the set of the natural numbers (for infinite sequence), or a set of the first $ n $ natural numbers. Each term in the sequence is called a term or an element or a member.
The $ {n^{th}} $ term is a formula with $ n $ in it which enables us to find any term of a sequence without having to go up from one term to the next. $ n $ stands for the term number so to find the $ {50^{th}} $ term we will just substitute $ 50 $ in the formula in the place of $ n $ .