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Chapter 7: Problem 62

Use technology to find a formula for the sum of the first \(n\) cubes \(1+8+27+\cdots+n^{3}\).

Short Answer

Expert verified
The formula to find the sum of the first $n$ cubes is: \[S_n = \frac{n^2(n + 1)^2}{4}\]

Step by step solution

01

Generate the first few sums of the series using technology.

We can use a CAS or a graphing calculator to generate the first few sums of the series. Here are the first few sums: Sum of first 1 cubes: 1 Sum of first 2 cubes: 9 Sum of first 3 cubes: 36 Sum of first 4 cubes: 100 Sum of first 5 cubes: 225 We can also write the sums as follows: 1鲁 = 1 1鲁 + 2鲁 = 9 1鲁 + 2鲁 + 3鲁 = 36 1鲁 + 2鲁 + 3鲁 + 4鲁 = 100 1鲁 + 2鲁 + 3鲁 + 4鲁 + 5鲁 = 225
02

Analyze the generated sums and search for a pattern.

Observe that the sums are all square numbers: 1, 9, 36, 100, and 225, which are the squares of the consecutive integers 1, 3, 6, 10, and 15. Notice that 1, 3, 6, 10, and 15 are triangular numbers, i.e., the sums of the first few consecutive integers: 1 = 1 3 = 1 + 2 6 = 1 + 2 + 3 10 = 1 + 2 + 3 + 4 15 = 1 + 2 + 3 + 4 + 5
03

Recognize the relationship between the sums and triangular numbers.

From our observations, we realize that the sum of the first n cubes is the square of the nth triangular number, T_n. So, S_n = T_n虏, where S_n represents the sum of the first n cubes, and T_n is the nth triangular number.
04

Define the nth triangular number and derive a formula for the sum of the first n cubes.

We know that the nth triangular number, T_n, is the sum of the first n consecutive integers. T_n = 1 + 2 + 3 + ... + n This sum can be expressed mathematically using the formula: \(T_n = \frac{n(n + 1)}{2}\) Now, we can use the relationship S_n = T_n虏 to find a formula for the sum of the first n cubes: S_n = \(\left(\frac{n(n + 1)}{2}\right)^2\) Therefore, the formula to find the sum of the first n cubes is: \[S_n = \frac{n^2(n + 1)^2}{4}\]

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