Problem On G.P –
For solving problems on G.P., there are some general instruction or information are given below –
It is always convenient to take –
a
(i) Three numbers in G.P. as ------, a, ar
r
(ii) a, ar, ar², ar³, ar⁴, ar⁵,…………….etc, when their product is not given.
-3 -5
Example.1) If the numbers, -------, x, ------- are in G.P.,
5 3
find the value of x
-3 -5
Ans.) Since, ------, x, and ------ are in G.P., we have
5 3
-5
(------)
x 3
------------ = -------------
-3 x
(------)
5
-5 -3
=> x² = (-------) X (-------)
3 5
=> x² = 1
=> x = ± 1
Hence, x = 1, or x = -1 (Ans.)
39
Example.2) The sum of three numbers in G.P. is ------- and their
10
product is 1. Find the numbers
a
Ans.) Let the required numbers be -----, a, ar. Then -
r
As per the given condition –
a
------ X a X ar = 1
r
=> a³ = 1 = 1³
=> a = 1
1
Putting the value of a = 1 in numbers in G.P. we find -----, 1, r
r
39
As per the given condition, the sum of these numbers is ------, and
10
1 39
------- + 1 + r = --------
r 10
1 + r + r² 39
=> ------------- = --------
r 10
=> 10 (1 + r + r²) = 39 r
=> 10 + 10r + 10r² = 39r
=> 10r²- 29r + 10 = 0
=> 10r²- (25 + 4)r + 10 = 0
=> 10r²- 25r – 4r + 10 = 0
=> 5r (2r – 5) – 2 (2r – 5) = 0
=> (2r – 5) (5r – 2) = 0
=> (2r – 5) = 0 and (5r – 2) = 0
=> r = 5/2 or r = 2/5
5 2 2 5
So,the required numbers are (------, 1, ------) or (------, 1, ------)
2 5 5 2
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