Question

The sum of four consecutive numbers in ap is 32 and the ratio of the products of the two middle terms is 7:15 find the number?

Answer 1

Let the four consecutive numbers in AP be (a - 3d), (a - d), (a + d) and (a + 3d)

So, according to the question.

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

4a = 32

a = 32/4

a = 8 ......(1)

Now, (a - 3d)(a + 3d)/(a - d)(a + d) = 7/15

15(a² - 9d²) = 7(a² - d²)

15a² - 135d² = 7a² - 7d²

15a² - 7a² = 135d² - 7d² 

8a² = 128d²

Putting the value of a = 8 in above we get.

8(8)² = 128d²

128d² = 512

d² = 512/128

d² = 4

d = 2

So, the four consecutive numbers are 

8 - (3*2)

8 - 6 = 2

8 - 2 = 6

8 + 2 = 10

8 + (3*2)

8 + 6 = 14

Four consecutive numbers are 2, 6, 10 and 14

 

Hope This Helps :)

Answer 2

Let the four consecutive numbers in AP be a-3d,a-d,a+d,a+3d.

So,

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

4a=32

a=8 1

(a-3d)(a=3d) / (a-d)(a+d)=7/15

15(a2-9d2) = 7(a2-d2)

15a2-135d2 = 7a2-7d2

8a2-128d2 = 0

d2 = 4, ±2

Therefore d=4 or d=±2 1mark

So, when a=8 and d=2, the numbers are 2,6,10,14.

When a=8,d=-2 the numbers are 14,10,6,2

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student-name Pooja Khanna answered this
508 helpful votes in Math, Class XII-Science
Here is the link for the answer to a similar query:
https://www.meritnation.com/ask-answer/question/the-sum-of-four-consecutive-numbers-in-an-ap-is-32-and-the-r/arithmetic-progressions/4211693
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student-name Karthika N J answered this
14 helpful votes in Math, Class XII-Science
Thank you for the answer. Thumps up for you.

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student-name Devanshi Modha answered this
47 helpful votes in Math, Class XII-Science
How do u get d^2

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student-name Raghuraman answered this
994 helpful votes in Math, Class XII-Science
Let the four consecutive numbers in AP be a-3d,a-d,a+d,a+3d.

So,
a-3d+a-d+a+d+a+3d=32
4a=32
a=8 1
(a-3d)(a=3d) / (a-d)(a+d)=7/15
15(a2-9d2) = 7(a2-d2)
15a2-135d2 = 7a2-7d2
8a2-128d2 = 0
d2 = 4, ±2
Therefore d=4 or d=±2 1mark
So, when a=8 and d=2, the numbers are 2,6,10,14.
When a=8,d=-2 the numbers are 14,10,6,2