Question

The sum of four consecutive numbers in an AP is 32 and the ratio of the product of the
first and the last terms to the product of the two middle terms in 7:15. Find the numbers.​

Answer 1

Step-by-step explanation:

$\huge\bf{\mid{\overline{\underline{ANSWER}\mid}}}$

Always remember:

For three terms in A. P =>

a-d, a, a+d

For four terms in A. P =>

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

$\huge\bold{Given:-} \\ \\ Sum = 32 \\ \\ Product \: \: of \: \: (a+3d)(a-3d) : (a-d) (a+d) = 7 : 15$

$a - \cancel{ 3d }+ a - \cancel{ d }+ a + \cancel{d} + a + \cancel{3d} = 32 \\ \\ 4a = 32 \\ \\ \boxed{a = 4} \\ \\ \implies \frac{(a - 3d)(a + 3d)}{(a - d)(a + d)} = \frac{7}{15} \\ \\ \implies \frac{ {a}^{2} - 9 {d}^{2} }{ {a}^{2} - {d}^{2} } = \frac{7}{15} \\ \\ \implies 15 {a}^{2} - 135 {d}^{2} = 7 {a}^{2} - 7 {d}^{2} \\ \\ \implies \cancel{8} {a}^{2} = \cancel{ 128} {d}^{2} \\ \\ \implies {a}^{2} = 16 {d}^{2} \\ \\ \implies 16 = 16 {d}^{2} \\ \\ \implies \boxed{d = 1}$

Number are,

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

4-3 , 4-1 , 4+1 ,4+3

1 , 3 , 5 , 7

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 .