Understanding the Missing Number in a Sequence: A Guide to Identifying Cubed Numbers
When facing a problem that involves identifying a missing number in a sequence, one effective approach is to look for patterns that the numbers in the sequence might follow. In the case of the sequence 1, 8, 27, 125, 216, the key is to identify the relationship between each term and the natural numbers they represent. In this article, we will explore how to find the missing number in the sequence, understand the pattern, and explain why the answer is 64.
Identifying the Pattern in Cubed Numbers
The sequence given is: 1, 8, 27, 125, 216. Let's break down each term by finding the cube of natural numbers.
Cubes of Natural Numbers:
13 1 23 8 33 27 43 64 53 125 63 216From the above list, we can see that the original sequence 1, 8, 27, 125, 216 is a part of the sequence of cubes of natural numbers. The missing number in the sequence is 64, which is the cube of 4.
Exploring the Pattern in Depth
The sequence can be represented as a series of cubed numbers:
13 1 23 8 33 27 43 64 53 125 63 216By carefully examining the sequence, we can see that 64 is the number that is missing in place of the sequence gap. This missing number is the cube of 4, reinforcing the sequence as 1, 8, 27, 64, 125, 216, and so on.
Generalizing the Cubic Sequence
The cubic sequence can be generalized as follows:
n3 the cube of the natural number nThe sequence of cubes of the first six natural numbers is as follows:
13 1 23 8 33 27 43 64 53 125 63 216Therefore, the missing number is 43 64, completing the sequence.
Common Mistakes and Clarifications
It's important to note that the missing number is not 343 (which would be 73), as the sequence is directly following the cubes of the natural numbers up to 6. There is no number between 3 and 4 in the natural numbers sequence, nor is there a missing number between 4 and 5 in terms of cubes.
The sequence 1, 8, 27, 64, 125, 216, 343... would only be correct if the question was for the seventh term, but since the missing number in this sequence is specifically in the gap between the fourth and fifth terms, the answer is unequivocally 64.