Mathematical Models

Problem Solving and Sequences

George Polya and How to Solve It

George Polya (1887–1985) was one of the most influential mathematicians of the twentieth century. His research contributions range from complex analysis, mathematical physics, probability theory, geometry, and combinatorics. In his book, How to Solve It, Polya reveals how the mathematical method of demonstrating a proof or finding an unknown can be of help in attacking any problem that can be “reasoned” out. His process includes four steps; 1) Understand the problem. 2) Devise a plan. 3) Carry out the plan. 4) Review/reflect.

Arithmetic and Geometric Sequences

Arithmetic Sequence

An arithmetic Sequence is a list of numbers in which consecutive numbers share a common difference.

Ex: 5, 10, 15, 20, 25,...

The common difference is 5.

Ex: 3, 6, 9, 12, 15,...

The common difference is 3.

Geometric Sequence

A geometric sequence is a list of numbers in which consecutive numbers share a common ratio. Each number after the first is calculated by multiplying the preceding number by the common ratio.

Ex: 1, 4, 16, 64, 256,...

The common ratio is 4.

Ex: 2, 4, 8, 16, 32,...

The common ratio is 2.

Fibonacci Sequence

The Fibonacci sequence is a series of numbers where a number is found by adding up the two numbers before it. Starting with 0 and 1, the sequence goes 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and so forth.

Named after Fibonacci, also known as Leonardo of Pisa or Leonardo Pisano, Fibonacci numbers were first introduced in his Liber abaci in 1202. The son of a Pisan merchant, Fibonacci traveled widely and traded extensively. Math was incredibly important to those in the trading industry and his passion for numbers was cultivated in his youth.

Knowledge of numbers is said to have first originated in the Hindu-Arabic arithmetic system, which Fibonacci studied while growing up in North Africa. Prior to the publication of Liber abaci, the Latin-speaking world had yet to be introduced to the decimal number system. He wrote many books about geometry, commercial arithmetic and irrational numbers. He also helped develop the concept of zero.

Arthur Benjamin: The magic of Fibonacci numbers
The Fibonacci Sequence: Nature's Code