What is the next number in the sequence?
$1, 8, 21, 64, 125$
Answer (2)
Analyze the given sequence and assume it follows a cubic polynomial form.
Set up a system of equations by plugging in n = 1 , 2 , 3 , 4 , 5 into the cubic polynomial.
Solve the system of equations to find the coefficients of the polynomial.
Calculate the next term in the sequence by plugging in n = 6 into the polynomial and round to the nearest whole number. The next number in the sequence is 221 .
Explanation
Problem Analysis We are given the sequence 1 , 8 , 21 , 64 , 125 and asked to find the next number. Let's analyze the sequence to identify a pattern. We can explore differences between consecutive terms and look for a polynomial that fits the sequence. We will assume the sequence follows a cubic polynomial form: f ( n ) = a n 3 + b n 2 + c n + d .
Setting up the Equations We can set up a system of equations by plugging in n = 1 , 2 , 3 , 4 , 5 into the cubic polynomial:
For n = 1 : f ( 1 ) = a ( 1 ) 3 + b ( 1 ) 2 + c ( 1 ) + d = a + b + c + d = 1 For n = 2 : f ( 2 ) = a ( 2 ) 3 + b ( 2 ) 2 + c ( 2 ) + d = 8 a + 4 b + 2 c + d = 8 For n = 3 : f ( 3 ) = a ( 3 ) 3 + b ( 3 ) 2 + c ( 3 ) + d = 27 a + 9 b + 3 c + d = 21 For n = 4 : f ( 4 ) = a ( 4 ) 3 + b ( 4 ) 2 + c ( 4 ) + d = 64 a + 16 b + 4 c + d = 64 For n = 5 : f ( 5 ) = a ( 5 ) 3 + b ( 5 ) 2 + c ( 5 ) + d = 125 a + 25 b + 5 c + d = 125
We have a system of 5 equations with 4 unknowns: a, b, c, and d.
Solving for Coefficients Solving this system of equations (using least squares method), we find the following values for the coefficients: a = 1 b = 0.85714286 c = − 5.14285714 $d = 4.8
Finding the Next Term Now we want to find the next number in the sequence, which corresponds to n = 6 . We plug n = 6 into the polynomial f ( n ) = a n 3 + b n 2 + c n + d :
f ( 6 ) = a ( 6 ) 3 + b ( 6 ) 2 + c ( 6 ) + d = 216 a + 36 b + 6 c + d
Substituting the values of a, b, c, and d: f ( 6 ) = 216 ( 1 ) + 36 ( 0.85714286 ) + 6 ( − 5.14285714 ) + 4.8 f ( 6 ) = 216 + 30.85714296 − 30.85714284 + 4.8 f ( 6 ) = 220.80000012
Final Answer Rounding the result to the nearest whole number, we get f ( 6 ) = 221 . Therefore, the next number in the sequence is 221.
Conclusion Thus, the next number in the sequence 1 , 8 , 21 , 64 , 125 is 221 .
Examples
Understanding number sequences is crucial in many fields, such as computer science, finance, and cryptography. For example, predicting stock prices often involves analyzing patterns in historical data, which can be modeled using mathematical sequences. Similarly, in cryptography, sequences are used to generate encryption keys and secure communication channels. By identifying patterns and predicting future values, we can make informed decisions and develop effective strategies in various real-world applications.
The next number in the sequence 1 , 8 , 21 , 64 , 125 is 216 , found by analyzing the pattern through differences and solving for a cubic polynomial. By substituting n = 6 into the polynomial, we derived this value. Thus, the sequence continues with the number 216 .
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