Question

*A number 36 is divided into four parts that are in A.P. such that the ratio of product of first and fourth part to the product of second and third part is 9 : 10. The lowest number of the A.P. is* fastest and correct answer will be brainest

Answer 1

The lowest number of AP is 6

Step-by-step explanation:

Let the 4 numbers in AP be

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

According to the question

$(a-3d) + (a-d) + (a+d) + (a+3d)=36$

$\implies 4a=36$

$\implies a=9$

And

$\frac{(a-3d)(a+3d)}{(a-d)(a+d)}=\frac{9}{10}$

$\implies \frac{a^2-9d^2}{a^2-d^2}=\frac{9}{10}$

$\implies \frac{9^2-9d^2}{9^2-d^2}=\frac{9}{10}$

$\implies 10(81-9d^2)=9(81-d^2)$

$\implies 10(9-d^2)=81-d^2$

$\implies 90-9d^2=81-d^2$

$\implies 8d^2=8$

$\implies d^2=1$

$\implies d=\pm 1$

If d = 1, the lowest number of AP is

$a-3d=9-3=6$

If d = -1, the lowest number of AP is

$a+3d=9+3(-1)=6$

Hope this answer is helpful.

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Answer 2

Step-by-step explanation:

Step-by-step explanation:

Let the 4 numbers in AP be

a-3d, a-d, a+d, a+3da−3d,a−d,a+d,a+3d

According to the question

(a-3d) + (a-d) + (a+d) + (a+3d)=36(a−3d)+(a−d)+(a+d)+(a+3d)=36

\implies 4a=36⟹4a=36

\implies a=9⟹a=9

And

\frac{(a-3d)(a+3d)}{(a-d)(a+d)}=\frac{9}{10}

(a−d)(a+d)

(a−3d)(a+3d)

=

10

9

\implies \frac{a^2-9d^2}{a^2-d^2}=\frac{9}{10}⟹

a

2

−d

2

a

2

−9d

2

=

10

9

\implies \frac{9^2-9d^2}{9^2-d^2}=\frac{9}{10}⟹

9

2

−d

2

9

2

−9d

2

=

10

9

\implies 10(81-9d^2)=9(81-d^2)⟹10(81−9d

2

)=9(81−d

2

)

\implies 10(9-d^2)=81-d^2⟹10(9−d

2

)=81−d

2

\implies 90-9d^2=81-d^2⟹90−9d

2

=81−d

2

\implies 8d^2=8⟹8d

2

=8

\implies d^2=1⟹d

2

=1

\implies d=\pm 1⟹d=±1

If d = 1, the lowest number of AP is

a-3d=9-3=6a−3d=9−3=6

If d = -1, the lowest number of AP is

a+3d=9+3(-1)=6a+3d=9+3(−1)=6

Hope this answer is helpful.