# CLASS-10GEOMETRIC PROGRESSION - PROBLEM ON G.P.

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