Question

the sum of four consecutive numbers in an AP is 32 and the ratio of the product of first and last tern to the product of the two middle term is 7:15 find the number

Answer 1

Answer:

2,6,10,14....

Step-by-step explanation:

Given The sum of four consecutive numbers in an AP is 32 and the ratio of the product of first and last term to the product of the two middle term is 7:15 to find the number

Let the four consecutive numbers be a, a + b,a + 2b, a + 3b

Their sum will be 32

So a + a + b + a + 2b + a + 3b = 32

          4a + 6b = 32

          2a + 3b = 16

          a = 16 - 3b / 2-----------(1)

 Given ratio is 7:15 for product of first and last term to product of middle term

 So a (a + 3b) / a + b)(a + 2b) = 7 / 15

 15a(a + 3b) = 7(a + b)(a + 2b)

 8a^2 + 24ab - 14b^2

4a^2 + 12ab - 7b^2

4a^2 - 2ab - 14ab - 7b^2

(2a - b)(2a - 7b) = 0

a = -7b/2, a = b/2

Equating with eqn 1 we get

-7b/2 = 16 - 3b / 2

b = - 4

b/2 = 16 - 3b/2

b = 4

So a = 16 - 3b / 2

    a = 16 - 3(4) / 2

  a = 2

Taking positive sign we get a = 2, b = 4

substituting in consecutive numbers we get

2, 2 + 4, 2 + 2(4), 2 + 3(4) and so on

2, 6, 10, 14

The numbers are 2,6,10,14

Answer 2

Answer:

 2 , 6, 10 , 14 .

Step-by-step explanation:

Let four consecutive number as

a - 3 d , a - d , a + d , a + 3 d .

Given their sum is 32 .

a + a + a + a + 3 d - 3 d = 32

4 a = 32

a = 8 .

Now :

Also given the ratio of the product of the first and the last term to product of two middle term to the product of two middle terms is 7:15 .

$\displaystyle{\dfrac{(a-3d)(a+3d)}{(a-d)(a+d)} = \dfrac{7}{15} }\\\\\\\displaystyle{\dfrac{a^2-(3d)^2}{a^2-d^2} = \dfrac{7}{15} }$

We have a = 8

$\displaystyle{\dfrac{64-(3d)^2}{64-d^2} = \dfrac{7}{15} }\\\\\\\displaystyle{\dfrac{64-9d^2}{64-d^2} = \dfrac{7}{15} }$

960 - 135 d² = 448 - 7 d²

135 d² - 7 d²  = 960 - 448

128 d²  = 512

d²  = 4

d = ± 2

When d = 2

Numbers are ,

a - 3 d , a - d ,  a + d , a + 2 d

= > 8 - 6 = 2

= > 8 - 2 = 6

= > 8 + 2 = 10

= > 8 + 6 = 14

When d = - 2

= > 8 + 6 = 14

= > 8 + 2 = 10

= > 8 - 2 = 6

= > 8 - 6 = 2 .

But both ways we get same numbers.

i.e.  2 , 6, 10 , 14 .