Question

There are 4 numbers in an A.P, the sum of 2 means is 18 and the product of the two extremes is 45. Find the numbers.

Answer 1

the 4 number is A.P is x-1,x x+1,x+2,

the sum of two mean is
x+1+x-1=18
the product of two extreme
(x-1)(x+2) = 45
x=9

Answer 2

Hey!

Let the AP in which there are four term be :-

( a - 3d ) , ( a - d ) , ( a + d ) , ( a + 3d )

• The sum of 2 means is 18

a - d + a + d = 18

2a = 18

a = 9

• Product of the extremes is 45

( a - 3d ) ( a + 3d ) = 45

[ Using identity :- ( x - y ) ( x + y ) = x² - y² ]

a² - ( 3d )² = 45

a² - 9d² = 45

9² - 9d² = 45

81 - 9d² = 45

- 9d² = 45 - 81

- 9d² = - 36

d² = 4

d = √4

d = ± 2


So, the AP's are :-

If a = 9 and d = 2

( a - 3d ) , ( a - d ) , ( a + d ) , ( a + 3d )

( 9 - 6 ) , ( 9 - 2 ) , ( 9 + 2 ) , ( 9 + 6 )

3 , 7 , 11 , 15

If a = 9 and d = -2


( a - 3d ) , ( a - d ) , ( a + d ) , ( a + 3d )

15 , 11 , 7 , 3