Geometric Progression

Consider the lists

(i) 30, 90, 270, 810, …..

(ii) 14, 116,164,1256

(iii) 30, 24, 19.2, 15.36, 12.288

Can we write the next term in each of the lists above?

In (i), each term is obtained by multiplying the preceding term by 3.

In (ii), each term is obtained by multiplying the preceding term by 14.

In (iii), each term is obtained by multiplying the preceding term by 0.8.

In all the lists given above, we see that successive terms are obtained by multiplying the preceding term by a fixed number.

Such a list of numbers is said to form Geometric Progression (GP).

This fixed number is called the common ratio ‘r’ of GP.

So in the above example (i), (ii), (iii) the common ratios are 3, 14, 0.8 respectively.

Let us denote the first term of a GP by a and common ratio r.

To get the second term according to the rule of Geometric Progression, we have to multiply the first term by the common ratio r, where a ≠ 0, r ≠ 0 and r ≠ 1

∴ The second term = ar

Third term = ar.r = ar²

∴ a, ar, ar^2 ..... is called the general form of a GP.

In the above GP the ratio between any term (except first term) and its preceding term is 'r' i.e., ara = ar2ar = ......... = r

If we denote the first term of GP by a₁, second term by a₂ ..... nth term by aₙ then a2a1 = a3a2 = ...... = anan−1 = r

∴ A list of numbers a1, a2, a3 .... an ... is called a geometric progression (GP), if each term is non zero and anan−1 = r (r ≠ 1)

where n is a natural number and n ≥ 2.

Some more examples of GP are:

(i) A person writes a letter to four of his friends. He asks each one of them to copy the letter and give it to four different persons with the same instructions so that they can move the chain ahead similarly. Assuming that the chain is not broken, the number of letters at first, second, third ... stages are

1, 4, 16, 64, 256 …………. respectively.

(ii) The total amount at the end of first, second, third …. year if ₹ 500/- is deposited in the bank with an annual rate of 10% interest compounded annually is

550, 605, 665.5 ……

(iii) A square is drawn by joining the midpoints of the sides of a given square. A third square is drawn inside the second square in the same way, and this process continues indefinitely. If a side of the first square is 16 cm, then the area of the first, second, third ….. square will be respectively.

256, 128, 64, 32, ….

(iv) Initially, a pendulum swings through an arc of 18 cm. On each successive swing, the length of the arc is 0.9th of the previous length. So the length of the arc at first, second, third....... swing will be respectively (in cm).

Now let us learn how to construct a GP. when the first term ‘a’ and common ratio ‘r’ are given. And also learn how to check whether the given list of numbers is a GP.