Chapter 26: Geometrical Progression

A series of quantities is said to be in Geometrical Progression (G.P.) when the ratio of any term to the one preceding it is constant. This constant multiplier is called the common ratio.

26.1 The General Term

If a is the first term and r is the common ratio, the series is represented as:

a, ar, ar², ar³, ...

To find the n^th term (l), we use the following formula:

l = ar^n-1

Example: Find the 6th term of the series 3, 6, 12, ...

Identify values: a = 3, r = 2, n = 6.
Apply formula: l = 3 × 2^6-1 = 3 × 2⁵
l = 3 × 32 = 96.

Result: The 6th term is 96.

26.2 Sum of a Geometrical Series

To find the sum (S) of the first n terms of a geometrical progression, we use the formula:

S = a(rⁿ - 1) / (r - 1)

If the ratio r is less than 1, it is often more convenient to use:

S = a(1 - rⁿ) / (1 - r)

26.3 Infinite Geometrical Series

When the common ratio r is a fraction less than 1, the terms of the series become smaller and smaller. As n increases infinitely, the sum of the series approaches a fixed limit:

S∞ = a / (1 - r)

Practical Application: Infinite series allow us to convert repeating decimals into fractions. For example, 0.333... is a G.P. where a = 3/10 and r = 1/10. Applying the formula gives exactly 1/3.

Doctor's Pro-Tip: Be very careful with negative ratios! A series with a negative r will oscillate between positive and negative values (e.g., 2, -4, 8, -16...).