What is the next number in the sequence?
$1, 8, 21, 64, 125$
Answer (2)
Analyze the given sequence 1 , 8 , 21 , 64 , 125 and observe that some terms are perfect cubes.
Identify the pattern as alternating between cubes and a number close to a square.
Determine that the next term in the sequence is a square number.
Conclude that the next square in the sequence is 1 8 2 = 324 .
Explanation
Problem Analysis We are given the sequence 1 , 8 , 21 , 64 , 125 and asked to find the next square number in the sequence. Let's analyze the sequence to identify a pattern.
Identifying Cubes Notice that some terms are perfect cubes: 1 = 1 3 , 8 = 2 3 , 64 = 4 3 , and 125 = 5 3 . Also, 64 = 8 2 is a perfect square. The number 21 is not a perfect cube or square. Let's look at the sequence of cubes: 1 3 = 1 , 2 3 = 8 , 3 3 = 27 , 4 3 = 64 , 5 3 = 125 , 6 3 = 216 .
Analyzing Differences Let's consider the possibility that the sequence alternates between cubes and values close to squares. The given sequence is 1 , 8 , 21 , 64 , 125 . We can rewrite it as 1 3 , 2 3 , 21 , 4 3 , 5 3 . If we look at the differences between consecutive terms, we have 8 − 1 = 7 , 21 − 8 = 13 , 64 − 21 = 43 , 125 − 64 = 61 .
Exploring Squares Let's consider the possibility that the sequence is related to n 3 − f ( n ) where f ( 1 ) = 0 , f ( 2 ) = 0 , f ( 3 ) = 6 , f ( 4 ) = 0 , f ( 5 ) = 0 . If the next term is a square, we can look at the sequence of squares: 1 , 4 , 9 , 16 , 25 , 36 , 49 , 64 , 81 , 100 , 121 , 144 , 169 , 196 , 225 , 256 , 289 , 324 , 361 , 400 , … .
Finding a Pattern The sequence could be constructed as follows: 1 3 = 1 , 2 3 = 8 , 3 2 + 12 = 21 , 4 3 = 64 , 5 3 = 125 . If we assume the next term is a square, we can look for a pattern. The cubes are 1 3 , 2 3 , 4 3 , 5 3 . The missing cube is 3 3 = 27 . The next cube is 6 3 = 216 . We are looking for a square. The sequence of bases is 1 , 2 , x , 4 , 5 , y . If we assume x = 3 and y = 6 , we can look for a pattern in the terms. The sequence is 1 3 , 2 3 , 21 , 4 3 , 5 3 , z 2 . We have 21 = 3 2 + 12 . Let's try z = 18 . Then 1 8 2 = 324 .
Identifying the Next Term The pattern could be: cube, cube, something, cube, cube, square. The sequence is 1 3 = 1 , 2 3 = 8 , 21 , 4 3 = 64 , 5 3 = 125 . The next term is 1 8 2 = 324 .
Final Answer Therefore, the next square in the sequence is 324 .
Examples
Consider a scenario where you are designing a pattern for a building's facade. The pattern involves alternating between cubic and square elements. The sequence 1 , 8 , 21 , 64 , 125 represents the sizes or dimensions of these elements. Finding the next square in the sequence, which is 324, helps you continue the pattern in a predictable and aesthetically pleasing manner. This ensures a harmonious design where the sizes of the elements follow a logical progression, making the facade visually appealing and structurally sound. The ability to predict the next element in such a sequence is crucial for maintaining consistency and balance in architectural designs.
The next number in the sequence 1 , 8 , 21 , 64 , 125 is determined by identifying a pattern involving perfect cubes and squares. After analyzing the sequence, it is concluded that the next number is 1 8 2 , which results in 324 . Hence, the next number in the sequence is oxed{324} .
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