Before we discuss Geometric Progression Formula, concepts, tricks etc, let us carry out an activity. Take a paper and fold it as many times as you can. So how many times did you fold the paper?

Maybe four to five times right? Now can you calculate the height of the stack of the paper after it has been folded number of times?

**How do you calculate it?** The answer to this is geometric progression. Let us study this in detail.

**Geometric Progression Concepts**

A **geometric progression **(GP) or **geometric sequence** is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the ** common ratio**. For example, sequence 2, 10, 50, 250, ... is a geometric progression with common ratio 5. Similarly, 20, 10, 5, 2.5, 1.25, ... is a geometric sequence with common ratio 1/2.

The behavior of a geometric sequence depends on the value of the common ratio. If the common ratio is:

- Positive, the terms will all be the same sign as the initial term.
- Negative, the terms will alternate between positive and negative.
- Greater than 1, there will be exponential growth towards positive or negative infinity (depending on the sign of the initial term).
- Between −1 and 1 but not zero, there will be exponential decay towards zero.
- −1, the progression is an alternating sequence
- Less than −1, for the absolute values there is exponential growth towards positive and negative infinity (due to the alternating sign).

**Geometric Progression Formulas**

Some of the important Geometric Progression formula are listed here for your reference.

- The general form of a Geometric Progression series is a, ar, ar
^{2}, ar^{3}and so on. - The nth term of a GP series is T
_{n}= ar^{n-1}

where a = first term and r = common ratio = T_{n}/T_{n-1}

- The sum of the first n terms of this geometric progression is

- The sum of infinite terms of a GP series S
_{∞ }= a/(1-r) where 0< r <1. - If a is the first term and r is the common ratio of a finite G.P which consists of m terms, then the nth term from the end will be = ar
^{m-n} - The nth term from the end of the G.P. with the last term l and common ratio r is l/(r
^{(n-1)}) - The geometric mean is defined as the
*n*th root (where n is the count of numbers) of the product of the numbers. Geometric mean b of two terms*a*and*c*is given by √(*ac*). If*a, b*and*c*are in geometric progression, then the ratio of the two consecutive terms should be equal.

b/a = c/b

Or, b² = ac

b = √(ac)

**Geometric Progression Problems**

Here are some problems based on the above discussed Geometric Progression formula.

**Question 1:** The number of bacteria in a certain culture doubles every hour. If there were 50 bacteria present in the culture originally, how many bacteria will be born in 12th hour?

**Solution**: The number of bacteria at any given time forms a GP whose terms are given by 50, 100, 200, ... where

a = 50, r = 100/50 = 2

Tn = arn -1

⇒ T12 = 50(2)12 - 1

= 50 x (2)11

= 102400

So, the number of bacteria born in 12th hour is 102400

**Question 2: **A ball is dropped from a height of 128 m. It bounces back rising to a height of 64 m. Each time it further touches the floor, it rises to the height of half the height it fell from before the previous bounce. Find the total distance traveled by the ball.

**Solution:** So, now total distance covered

= Distance traveled first time+2×{a÷(1-r)}

Where a=128 and r=½

=128+{128÷(1–1/2)}

=128+{128÷(1/2)}

=128+{128×2}

=128+256

=384