divide 56 into 4 parts which are in AP such that the ratio of product

1.	divide 56 into 4 parts which are in AP such that the ratio of product of extremes to the product of means is 5:6.
Answer» Answer: AP formed on dividing 56 into 4 parts will be 8 , 12 , 16 , 20 Step-by-step explanation: Let, 56 is DIVIDED in 4 parts i.e, a - 3 d , a - d , a + d , a + 3 d which are in AP so, → a - 3 d + a - d + a + d + a + 3 d = 56 → 4 a = 56 → a = 56 / 4 → a = 14 Now, as GIVEN that ratio of product of extremes and means is 5 : 6 therefore, → [(a - 3 d)(a + 3 d)] / (a - d)(a + d) = 5 / 6 → (a² - 9 d²) / (a² - d²) = 5 / 6 ( cross multiplying ) → 6 a² - 54 d² = 5 a² - 5 d² → a² - 49 d² = 0 ( putting a = 14 ) → (14)² - 49 d² = 0 → 196 = 49 d² → d² = 196 / 49 → d = ± 2 Therefore, on putting d = 2 FOUR parts will be → a - 3 d = 14 - 3 ( 2 ) = 8 → a - d = 14 - 2 = 12 → a + d = 14 + 2 = 16 → a + 3 d = 14 + 3 (2) = 20 (on taking d = -2 we will get the same A.P.)

divide 56 into 4 parts which are in AP such that the ratio of product of extremes to the product of means is 5:6.

Answer»

Answer:

AP formed on dividing 56 into 4 parts will be

8 , 12 , 16 , 20

Step-by-step explanation:

Let, 56 is DIVIDED in 4 parts i.e,

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

which are in AP

so,

→ a - 3 d + a - d + a + d + a + 3 d = 56

→ 4 a = 56

→ a = 56 / 4

→ a = 14

Now,

as GIVEN that ratio of product of extremes and means is 5 : 6

therefore,

→ [(a - 3 d)(a + 3 d)] / (a - d)(a + d) = 5 / 6

→ (a² - 9 d²) / (a² - d²) = 5 / 6

( cross multiplying )

→ 6 a² - 54 d² = 5 a² - 5 d²

→ a² - 49 d² = 0

( putting a = 14 )

→ (14)² - 49 d² = 0

→ 196 = 49 d²

→ d² = 196 / 49

→ d = ± 2

Therefore,

on putting d = 2

FOUR parts will be

→ a - 3 d = 14 - 3 ( 2 ) = 8

→ a - d = 14 - 2 = 12

→ a + d = 14 + 2 = 16

→ a + 3 d = 14 + 3 (2) = 20

(on taking d = -2 we will get the same A.P.)