Chapter 25: Arithmetical Progression

A series of quantities is said to be in Arithmetical Progression (A.P.) when the difference between any term and the one preceding it is constant. This constant difference is called the common difference.

25.1 The General Term

If a is the first term and d is the common difference, the series looks like this:

a, (a + d), (a + 2d), (a + 3d), ...

To find the n^th term (represented by l for "last term"), we use the formula:

l = a + (n - 1)d

Example: Find the 10^th term of the series 2, 5, 8, ...

Identify values: a = 2, d = 3, n = 10.
Apply formula: l = 2 + (10 - 1)3
l = 2 + (9)3 = 2 + 27 = 29.

Result: The 10^th term is 29.

25.2 Sum of an Arithmetical Series

To find the sum (S) of the first n terms of an arithmetical progression, we use the average of the first and last terms multiplied by the number of terms:

S = (n/2)(a + l)

Substituting the formula for l, we get the alternative version:

S = (n/2)[2a + (n - 1)d]

25.3 Arithmetical Means

When three quantities are in A.P., the middle one is called the Arithmetical Mean between the other two. To find the mean between any two numbers, simply find their average: (a + b) / 2.

Historical Note: The legend goes that the young mathematician Gauss discovered the sum formula as a child when his teacher asked the class to add all numbers from 1 to 100 to keep them busy. He realized 1+100=101, 2+99=101, and so on, reaching the answer in seconds.

Doctor's Pro-Tip: Always check the sign of your common difference (d). If the series is decreasing (e.g., 10, 7, 4...), d must be a negative number in your calculations!