Problem 20 - University Calculus: Early Transcendentals

Question

Evaluate the sums. a. \(\sum_{k=1}^{13} k\) b. \(\sum_{k=1}^{13} k^{2}\) c. \(\sum_{k=1}^{13} k^{3}\)

Step-by-Step Solution

Answer

a. 91, b. 819, c. 8281.

Step 1: Understand the Problem

We need to evaluate three different sums with given limits and expressions: 1. The sum of the first 13 natural numbers, 2. The sum of squares of the first 13 natural numbers, 3. The sum of cubes of the first 13 natural numbers.

Step 1: Calculating Sum of Natural Numbers

The sum of the first n natural numbers is given by the formula: \[ \sum_{k=1}^{n} k = \frac{n(n+1)}{2} \]For n = 13, the sum is: \[ \sum_{k=1}^{13} k = \frac{13(13+1)}{2} = \frac{13 \times 14}{2} = 91 \]

Step 2: Calculating Sum of Squares

The formula for the sum of squares of the first n natural numbers is: \[ \sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6} \]For n = 13, the sum of squares is: \[ \sum_{k=1}^{13} k^2 = \frac{13(13+1)(2 \times 13 + 1)}{6} = \frac{13 \times 14 \times 27}{6} = 819 \]

Step 3: Calculating Sum of Cubes

The sum of cubes of the first n natural numbers can be calculated using the formula: \[ \sum_{k=1}^{n} k^3 = \left(\frac{n(n+1)}{2}\right)^2 \]For n = 13, the sum of cubes is: \[ \sum_{k=1}^{13} k^3 = \left(\frac{13 \times 14}{2}\right)^2 = 91^2 = 8281 \]

Key Concepts

Natural Numbers

Natural numbers are the set of positive integers starting from 1 and going upwards like 1, 2, 3, 4, and so on. These numbers are the foundation of mathematics and are used extensively in counting and ordering. Natural numbers do not include zero, negative numbers, fractions, or decimals.

The sum of natural numbers is an arithmetic series. When calculating the sum of the first 'n' natural numbers, the formula that comes in handy is \( \sum_{k=1}^{n} k = \frac{n(n+1)}{2} \). This expression is derived considering the incremental addition from 1 up to 'n'.

For example, suppose we want to find the sum of the first 13 natural numbers. You simply plug '13' in as 'n' in the formula to get \( \sum_{k=1}^{13} k = \frac{13 \times 14}{2} = 91 \).

• This formula showcases that adding consecutive numbers forms a predictable pattern.
• The simplicity of this formula is rooted in its geometrical analysis, representing a straightforward increase.

Understanding natural numbers and their sums helps lay the groundwork for more complex mathematical operations.

Sum of Squares

The sum of squares of the first 'n' natural numbers is a bit more intricate than just summing the numbers themselves. When we square numbers and sum them, we highlight the cumulative area of each number on the number line.

The formula for calculating the sum of squares is \( \sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6} \). This formula emerges from an advanced derivation involving both arithmetic and geometry to account for each squared unit.

For instance, to find the sum of the squares of the first 13 natural numbers, substitute '13' for 'n' to get:

• \( \sum_{k=1}^{13} k^2 = \frac{13 \times 14 \times 27}{6} = 819 \)

This formula is essential when dealing with quadratic equations or when analyzing variance in statistics.

Understanding the sum of squares assists in various fields, including physics and engineering, where quadratic growth patterns are analyzed.

Sum of Cubes

The concept of the sum of cubes takes this exploration one dimension further. If sum of squares can relate to areas, the sum of cubes bears relation to volumes.

The sum of cubes of the first 'n' natural numbers uses a beautiful and elegant formula: \( \sum_{k=1}^{n} k^3 = \left(\frac{n(n+1)}{2}\right)^2 \). It shows a perfect symmetry as it ties back to the sum of natural numbers and forms a perfect square.

For example, when calculating the sum of the cubes for the first 13 natural numbers:

• \( \sum_{k=1}^{13} k^3 = (\frac{13 \times 14}{2})^2 = 91^2 = 8281 \)

This concept reveals the spatial increment visualization of cubes in three-dimensional space.

The sum of cubes is often encountered in more advanced fields of calculus and algebra, particularly while dealing with polynomial identities and theoretical physics.