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A GP is a sequence where you start with any object (number/function) and then keep on boringly multiplying by any other object (number/function) you like.
Example 18
Start with a number like
and keep on multiplying by a different number like 
.G.P.:
Example 19
Start with the function
say and keep on multiplying by the number 5 say.G.P.:
(b) The General GP
Start with a and keep on multiplying by the number r to generate the GP:
Notes:
(i) u1 = a = the first term
u2 = ar = the second term
u3 = ar2 = the third term etc..
And so u45 = ar44
Trick: If the forty-third term of a GP is 132 then u43 = 132 = ar42
(ii) The General Term is un = arn-1
(iii) The quotient r of any term and the previous one is a constant known as the common ratio. This is the test for a GP.
Test: A sequence is a GP if
= constant = r for all 
(c) The Sum Sn of a GP.
The formula for adding up the first n terms of a GP is given by:
The proof is not required.
(d) The Sum to Infinity of a GP
The formula for adding up an infinite GP is given by
.It is a special case of Sn with n = and only works iff -1 < r < 1.
(e) Tips
(i) If you know the first term u1 = a of a GP and the common ratio r you can find everything else.
(ii) Make sure you know what all the symbols mean.
u1 = a = the value of the first term
r = common ratio = any term/previous one
un = value of the object in the nth place
n = the place in the list
Sn = the sum of the first n terms
= the sum to infinity(f) Tricks for 3 consecutive terms in a GP.
(i) If you are asked to choose 3 consecutive numbers in a GP choose them as:
(ii) If you are told that a, c, b are consecutively in a GP then simply write:
(iii) The Geometric Mean (GM)
Given 2 numbers a, b can you find a number between these so that all three are consecutively in a GP. From (ii) the GM =
Example 20
The third term of a GP is 36 and the sixth term is 4.5. Find the first 4 terms.Solution
Example 21
Which term is 160 in the GP: 5, 10, 20, ....Solution
Example 22
If 2, x, 8, y are four consecutive terms of a GP find x and y.Solution
Example 23
How many terms of the GP: 8, 24, 72,.... must be added to give 968?Solution
Example 24
Find the sum to infinity of the GP: 1, cos2x, cos4x .....when x = 30oSolution
Example 25
Find 3 consecutive numbers in a GP whose sum is 38 and whose product is 1728.Solution
Example 26
The sum to infinity of the GP: a, ar, ar2.... is 15. The sum to infinity of the GP: a2, a2r2, a2r4,..... is 45. Find a and r.Solution
