Number theory is the branch of pure math- ematics that studies the properties of the integers. Thinking about properties of integers can be rec- reational as well as performed by mathematicians. In the past, a common quest was to find formulas to calculate the sums of powers of integers. Be- cause there were no computers to calculate these sums of powers, number theorists developed for- mulas to find sums of powers of integers indirectly. Bernoulli numbers were developed by Swiss math- ematician Jakob Bernoulli in the late 17th century. These numbers were used in a complex formula to calculate the sums of powers of integers quickly well, relatively quickly. It was still a labor-intensive calculation. Today, computers can be used to solve this problem. A computer can calculate the sums of powers of integers directly and almost instanta- neously without Bernoulli numbers. Examples of this task are shown below. Example 1 This example shows the calculation of the sum of the first five powers of 3 directly. Powers of 3 Sum of Powers of 3 30 = 1 1 31 = 3 4 32 = 9 13 33 = 27 40 34 = 81 121 Example 2 This example shows the calculation of the sum of the first four powers of 10. Powers of 10 Sum of Powers of 10 100 = 1 1 101 = 10 11 102 = 100 111 103 = 1000 1111 Assignment 1 Apply what you have learned about computa- tional thinking to write an algorithm for calculat- ing the sums of powers of integers. Allow input of any integer. Calculate the sum of the first 100 powers of that integer. Follow the four actions of computational thinking. Assignment 2 Study Example 2 above for calculating the sums of the powers of 10. Look for a pattern in the sums. Write a shortcut algorithm to find the sum of any number of powers of 10. Follow the four ac- tions of computational thinking. Math and Java Sums of Powers Algorithm Chapter 1 Computational Thinking 11 Copyright Goodheart-Willcox Co., Inc.