Question

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 1

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.)

Answer 2

$\bigstar$ $\sf{\green{\underline{\underline{\orange{To\:Find:-}}}}}$

• The four parts number of 56 which are in A.P .

$\bigstar$ $\sf{\purple{\underline{\underline{\blue{SOLUTION:-}}}}}$

☞ GIVEN :-

• Divide 56 into 4 parts which are in A.P .

• And the Ratio of product of extremes to the product of means is “5:6” .

☞ CALCULATION :-

☞ Let the four parts be

1. (a - 3d)
2. (a - d)
3. (a + d)
4. (a + 3d)

Where,

• a = first term .

• d = common difference .

☞ According to question, the sum is ‘56’

$\tt{\implies\:(a\:-\:3d)\:+\:(a\:-\:d)\:+\:(a\:+\:d)\:+\:(a\:+\:3d)\:\:=\:\:56\:}$

$\tt{\implies\:4a\:=\:56\:}$

$\tt{\red{\boxed{\implies\:a\:=\:14\:}}}$

☞ Also given that,

$\tt{\purple{\boxed{{\dfrac{(a\:-\:3d)\:(a\:+\:3d)}{(a\:-\:d)\:(a\:+\:d)}}\:\:=\:\:{\dfrac{5}{6}}\:}}}$

$\tt{\implies\:{\dfrac{(a^2\:-\:9d^2)}{(a^2\:-\:d^2)}}\:=\:{\dfrac{5}{6}}\:}$

☞ Putting the value of “a = 14” in the above equation, we get

$\tt{\implies\:{\dfrac{(14)^2\:-\:9d^2)}{(14)^2\:-\:d^2}}\:=\:{\dfrac{5}{6}}\:}$

$\tt{\implies\:{\dfrac{196\:-\:9d^2}{196\:-\:d^2}}\:=\:{\dfrac{5}{6}}\:}$

$\tt{\implies\:1176\:-\:54d^2\:=\:980\:-\:5d^2\:}$

$\tt{\implies\:49d^2\:=\:196\:}$

$\tt{\implies\:d^2\:=\:4\:}$

$\tt{\red{\boxed{\implies\:d\:=\:±2\:}}}$

☞ Now, putting

• “a = 14” and “d = +2”

1. (a - 3d) = (14 - 6) = 8
2. (a - d) = (14 - 2) = 12
3. (a + d) = (14 + 2) = 16
4. (a + 3d) = (14 + 6) = 20

☞ If we take “a = 14” and “d = (-2)”, then same four values is resulting as the above four values .

☞ Thus, the four parts are “8” , “12” , “16” , “20” .