Arithmetic Progression
In an arithmetic progression, the difference between consecutive terms is a constant d, and the first term is a. The n-th term is a plus n minus 1 times d. The sum of the first n terms is n over 2 times 2a plus n minus 1 times d, which also equals n over 2 times the first plus the last term. The number of terms is the last minus the first, over d, plus 1. The arithmetic mean of a and b is a plus b over 2.
Geometric Progression
In a geometric progression, the ratio of consecutive terms is constant, r. The n-th term is a times r to the n minus 1. The sum of the first n terms is a times r to the n minus 1 over r minus 1 when r is greater than one, and a times 1 minus r to the n over 1 minus r when r is less than one. The sum to infinity when the absolute value of r is less than one is a over 1 minus r. The geometric mean of a and b is the square root of ab.
Harmonic Mean
The harmonic mean of a and b is 2ab over a plus b. The arithmetic mean is at least the geometric mean, which is at least the harmonic mean, and the geometric mean squared equals the arithmetic mean times the harmonic mean.
Reading Number Series
Look at differences for an arithmetic pattern and ratios for a geometric pattern. Check for squares and cubes. If the differences themselves form a pattern, as in 2, 6, 12, 20, 30 with differences 4, 6, 8, 10, extend that pattern. Check alternate terms separately.
Shortcuts
The sum of an arithmetic progression equals the number of terms times the average of the first and last terms. One plus two plus up to 100 is 5050. The sum 1 plus 2 plus 4 up to 2 to the n minus 1 equals 2 to the n minus 1.
Worked Examples
Example 1: Find the 20th term of 3, 7, 11 and so on.
Solution: The first term is 3 and d is 4, so the term is 3 plus 19 times 4, which is 79.
Example 2: Find the sum to infinity of 8, 4, 2 and so on.
Solution: The first term is 8 and r is one half, so the sum is 8 over one half, which is 16.
Example 3: Find the next number in 2, 6, 12, 20, 30.
Solution: The differences are 4, 6, 8, 10, so the next difference is 12 and the next number is 42.
Quick Check
Question 1: What is the sum of the first 100 natural numbers?
Answer: 5050.
Question 2: What is the sum to infinity of a geometric series?
Answer: a over 1 minus r, when the absolute value of r is less than 1.
Question 3: What relation holds between AM, GM and HM?
Answer: GM squared equals AM times HM.