Problem 79 - Algebra and Trigonometry

Question

Find the sum of each sequence. \(\sum_{k=5}^{20} k^{3}\)

Step-by-Step Solution

Answer

The sum of the sequence \(\sum_{k=5}^{20} k^{3}\) is 45700.

Step 1 - Understand the Summation Notation

The given exercise asks to find the sum of the cubes of integers from 5 to 20. The notation \(\sum_{k=5}^{20} k^{3}\) represents the sum of the sequence \(5^{3}, 6^{3}, 7^{3}, \dots, 20^{3}\).

Step 2 - Identify the Limits of Summation

The lower limit of summation is 5 and the upper limit is 20. So, calculate the cubes of each integer from 5 to 20.

Step 3 - Calculate Each Term in the Sequence

Calculate the cubes of the numbers from 5 to 20: \(5^{3} = 125\), \(6^{3} = 216\), \(7^{3} = 343\), \(8^{3} = 512\), \(9^{3} = 729\), \(10^{3} = 1000\), \(11^{3} = 1331\), \(12^{3} = 1728\), \(13^{3} = 2197\), \(14^{3} = 2744\), \(15^{3} = 3375\), \(16^{3} = 4096\), \(17^{3} = 4913\), \(18^{3} = 5832\), \(19^{3} = 6859\), \(20^{3} = 8000\).

Step 4 - Sum All Calculated Terms

Add all the calculated terms: \(125 + 216 + 343 + 512 + 729 + 1000 + 1331 + 1728 + 2197 + 2744 + 3375 + 4096 + 4913 + 5832 + 6859 + 8000\).

Step 5 - Provide the Final Sum

The final sum of the sequence is: 45700.

Key Concepts

Sequences

A sequence is a list of numbers arranged in a specific order. Each number in a sequence is referred to as a term. Sequences can be finite or infinite. In this exercise, we deal with a finite sequence which starts at 5 and ends at 20. This sequence includes all integers between 5 and 20, inclusive.
Sequences often follow a specific pattern, such as the arithmetic or geometric patterns. In this problem, each term in the sequence is a cube of an integer from 5 to 20. The terms of this sequence are:
• 5³=125
• 6³=216
• 7³=343
• 8³=512
Understanding sequences helps in identifying patterns and calculating terms in a systematic manner.

Series

A series is the sum of the terms of a sequence. When the terms of a sequence are added together, you get a series. If a sequence is given by \{a_n\}, the corresponding series is represented as \(\sum a_n\).
In our example, we consider the series derived from the sequence of cubes of integers from 5 to 20. The notation \(\sum_{k=5}^{20} k^{3}\) asks us to find the sum of the cubes of these numbers.
A series can be finite, like in our case, or infinite if it goes on without stopping. For a finite series, we can simply add all the terms together to get the sum. The process often involves breaking down the problem into manageable steps, like calculating each term and then summing them up systematically.

Summation

Summation notation is a concise way of expressing the sum of a series. This notation uses the Greek letter sigma (\(\sum\)) to represent the sum. The general form involves an index of summation, upper and lower limits, and the terms to be summed.
In the given exercise, the summation notation \(\sum_{k=5}^{20} k^{3}\) tells us to sum the cubes of all integers from 5 to 20. The index of summation is \(k\), the lower limit is 5, and the upper limit is 20.
Summation helps in structuring and simplifying mathematical expressions where adding multiple terms is required. It's particularly useful in calculus, statistics, and various areas of algebra. For our sequence, the step-by-step approach using summation notation provided a clear method to reach the final sum.