Question

In an A.P sum of first four terms is 10, the product of the 1st and 4th term is 10 times the product of 2nd and 3rd term. Find that numbers

Answer 1

so four numbers are -20,-5,20,25

Answer 2

Answer:

Numbers are -20 , -5 , 10 , 25.

Step-by-step explanation:

Given: Sum of four terms in AP = 10

           Product of 1st & 4th term = 10 times product of 2nd & 3rd term

To find: 4 Terms of AP

Let 4 terms of AP are   a-3d , a-d , a+d , a+3d

1st term, a = a - 3d

2nd term,  $a_2$ = a - d

3rd term,  $a_3$  = a + d

4th term,  $a_4$   = a + 3d

According to question,

$a+a_2+a_3+a_4=10$

(a-3d) + ( a-d ) + ( a+d ) + ( a+3d ) = 10

a - 3d + a - d + a + d + a + 3d = 10

4a = 10

2a = 5

$a=\frac{5}{2}$

a = 2.5

$a\times a_4=10\times a_2\times a_3$

(a-3d) × ( a+3d ) = 10 × ( a-d ) × ( a+d )

a² - 9d² = 10 × ( a² - d² )

a² - 9d² =  10a² - 10d²

10d² - 9d² = 10a² - a²

d² = 9a²

d = ± 3a

d = ± 3 × 2.5

d = ± 7.5

when d = 7.5

1st term, a = 2.5 - 3×7.5 = -20

2nd term,  $a_2$ = 2.5 - 7.5 = -5

3rd term,  $a_3$  = 2.5 + 7.5 = 10

4th term,  $a_4$   = 2.5 + 3×7.5 = 25

when d = -7.5

1st term, a = 2.5 - 3×(-7.5) = 25

2nd term,  $a_2$ = 2.5 - (-7.5) = 10

3rd term,  $a_3$  = 2.5 + (-7.5) = -5

4th term,  $a_4$   = 2.5 + 3×(-7.5) = -20

in both case nos are same.

Theerfore, Numbers are -20 , -5 , 10 , 25.