- A series in which each preceding term is formed by multiplying it by a constant factor is called a Geometric Progression G. P. The constant factor is called the common ratio and is formed by dividing any term by the term which precedes it.
- In other words, a sequence, is called a geometric progression
- If = constant for all n ∈ N.
- The General form of a G. P. with n terms is a, ar, ar2,…arn -1
Thus if a = the first term
r = the common ratio
Tn = nth term and
Sn = sum of n terms
- General term of GP =
Example.1 Find the 9th term and the general term of the progression.
- The given sequence is clearly a G. P. with first term a = 1 and common ratio = r =
Try yourself:How many terms are there in the GP 5, 20, 80, 320, ..........., 20480?
➢ Sum of n terms of a G.P
➢ Sum of Infinite G.P
If a G.P. has infinite terms and -1 < r < 1 or |x| < 1, then sum of infinite G.P is
Example.2 The inventor of the chess board suggested a reward of one grain of wheat for the first square, 2 grains for the second, 4 grains for the third and so on, doubling the number of the grains for subsequent squares. How many grains would have to be given to inventor? (There are 64 squares in the chess board).
- Required number of grains = 1 + 2 + 22 + 23 + ……. To 64 terms =
➢ Recurring Decimals as Fractions
- If in the decimal representation a number occurs again and again, then we place a dot (.) on the number and read it as that the number is recurring.
- Example: 0.5 (read as decimal 5 recurring).
This mean 0. = 0.55555…….∞
0. 4 = 0.477777……∞
These can be converted into fractions as shown in the example given below
Example.3 Find the value in fractions which is same as of
➢ Properties of G.P.
- If each term of a GP is multiplied or divided by the same non-zero quantity, then the resulting sequence is also a GP.
Example: For G.P. is 2, 4, 8, 16, 32…
- Selection of terms in G.P.
Sometimes it is required to select a finite number of terms in G.P. It is always convenient if we select the terms in the following manner:
If the product of the numbers is not given, then the numbers are taken as a, ar, ar2, ar3, ….
- Three non-zero numbers a, b, c are in G.P. if and only if
b2 = ac or b is called the geometric mean of a & c
- In a GP, the product of terms equidistant from the beginning and end is always same and equal to the product of first and last terms as shown in the next example.