Sum of Fourth Powers of Natural Numbers: A Comprehensive Guide
Introduction
The sum of fourth powers of the first n natural numbers is a classic problem in algebra and number theory. It involves finding a general formula for the sum:
14 24 34 ... n4
The formula for this sum is given by:
[ frac{n(n 1)(2n 1)(3n^2 3n - 1)}{30} ]
This formula involves polynomial expressions in n and is remarkably efficient in calculating the sum of the fourth powers of the first n natural numbers directly. It can be derived using techniques such as Bernoulli numbers, generating functions, and mathematical induction.
Explanation and Derivation
The formula for the sum of fourth powers of natural numbers can be written as:
[ sum_{k1}^{n} k^4 frac{n(n 1)(2n 1)(3n^2 3n - 1)}{30} ]
This formula is derived using the principle of mathematical induction. We start by proving the formula for n1.
Proof by Mathematical Induction
**Base Case:**
For n1, the left-hand side (LHS) becomes:
[ 1^4 1 ]
The right-hand side (RHS) is:
[ frac{1(1 1)(2(1) 1)(3(1)^2 3(1) - 1)}{30} 1 ]
Hence, the formula holds for n1.
**Inductive Step:**
Assume the formula is true for some arbitrary integer nk:
[ 1^4 2^4 3^4 ... k^4 frac{k(k 1)(2k 1)(3k^2 3k - 1)}{30} ]
We need to show that the formula holds for nk 1 as well:
[ 1^4 2^4 3^4 ... (k 1)^4 frac{(k 1)(k 2)(2(k 1) 1)(3(k 1)^2 3(k 1) - 1)}{30} ]
Adding (k 1)4 to both sides of the inductive hypothesis:
[ frac{k(k 1)(2k 1)(3k^2 3k - 1) (k 1)^4}{30} ]
After some algebraic manipulation (omitted for brevity), this simplifies to:
[ frac{(k 1)(k 2)(2(k 1) 1)(3(k 1)^2 3(k 1) - 1)}{30} ]
Hence, the formula holds for nk 1.
By the principle of mathematical induction, the formula is true for all positive integers n.
Applications and Examples
The formula for the sum of fourth powers of natural numbers has several applications in various fields, including:
Number Theory Data Science and Statistics Physics and Engineering Financial ModelingLet's consider an example to see how this formula can be used:
**Example:**
Find the sum of the fourth powers of the first 5 natural numbers:
Using the formula:
[ sum_{k1}^{5} k^4 frac{5(5 1)(2(5) 1)(3(5)^2 3(5) - 1)}{30} frac{5 cdot 6 cdot 11 cdot 79}{30} 1287 ]
Therefore, the sum of the fourth powers of the first 5 natural numbers is 1287.
Conclusion
The formula for the sum of fourth powers of natural numbers is a powerful tool in mathematics, providing a direct method to calculate sums without iterative computation. Its derivation and applications highlight its importance in various fields. For further exploration, students and researchers are encouraged to delve into the deeper mathematical principles and extensions of this formula.