PREFACE SCHOOL PROGRAMS / PROGRAMAS FOR THE PUBLIC ANCIENT COUNTING METHOD PRIME NUMBER NUMERAL SYSTEM GOLDEN RATIO PROBABILITY WEIGHTS AND MEASUREMENTS SEQUENCE AND SERIES GEOMETRIC FIGURE FOOTBALL SYMMETRY FRACTAL EASTERN MATHEMATICIANS WESTERN MATHEMATICIANS CRYPTOLOGY MATHEMATICS AND ARCHITECTURE MATHEMATICS AND NATURE MATHEMATICS AND FINE ARTS PROTRACTOR TOWER OF HANOI ZHUSUAN AND ABACUS

SEQUENCE AND SERIES

A sequence is a list of numbers that are in order. In an Arithmetic Sequence, the difference between one term and the next is a constant, called Common Difference. If there has only three numbers (a, b and c) in an arithmetic sequence, b is the Arithmetic Mean of the sequence. In a Geometric Sequence, the same ratio between each term is in equal, it is called the Common Ratio. For instance, if the sequence consists of a, b and c, b is the Geometric Mean of the sequence.

The sum of the numbers in a sequence (a₁ + a₂ + ... .. + a_n) is called a Series. Therefore, when a₁ + a₂ + ... .. + a_n is an arithmetic sequence, its series is known as Arithmetic Series. In the same way, if the sequence is a geometric sequence, its series is called Geometric Series.

Series on the Chessboard

It was said that in the ancient times, there was a king playing chess with his minister. The king asked what kind of reward the minister wanted. The minister replied, "I only want to have one grain of rice placed on the first square, two grains on the second, four grains on the third, eight grains on the fourth and so on, until all squares on the chessboard are filled with grains." The king accepted his request instantly as he thought this was very easy.

Let us calculate how many grains of rice the king had to give the minister!

1 grain on the first square means 2⁰=1; 2 grains on the second square means 2¹=2; 4 grains on the third square means 2²=4. Continuing with this, the 64th square needs 2⁶³ grains, which is equal to:

From the king's story, it can be seen that the increment in the values of geometric series can be prodigious because the ratio of each successive term is a constant.

According to the weight of rice in general, 600 grains weigh about 50g. In other words, the king has to give approximately 15,372 tons of rice to the minister. If a person was able to count two grains per second, he could finish counting this amount of rice in 292,500,000,000 years even if he works day and night without sleep. According to the statistics, the world's population in 2016 is already over 7,300 billion. If everyone in the world all count the grains non-stop without sleep, it will take about 40 years to finish counting this amount of rice!

Did you know?

1. Chess is a two-player strategy board game played on a chessboard. It first appeared in India and until the late 15th century, the rules of modern chess were formed. The World Chess Federation (FIDE) was set up in 1924. It is responsible for governing international chess competitions, calculating the Elo ratings of competitors, awarding the FIDE Master, International Master, Grandmaster and women's FIDE titles to players.
2. Compound interest is the addition of interest to the principal sum of a loan or deposit. The higher the interest rate and the shorter the period, the higher the return.