find four natural numbers in ap such that their sum is 20 and product is 384
Hi Sadia Hafeez,
Kindly find the solution to your query:
Let First term of A.P. = a
And
Common difference = 2d
So
Four numbers in A.P. = ( a - 3d ) , ( a - d ) , ( a + d ) and ( a + 3d )
So, given
⇒ ( a - 3d ) + ( a - d ) + ( a + d ) + ( a + 3d ) = 20
⇒ 4a = 20
⇒ a = 5
And
⇒ ( a - 3d ) ( a - d ) ( a + d ) ( a + 3d ) = 384
⇒ ( a - 3d ) ( a + 3d ) ( a - d ) ( a + d ) = 384
⇒ ( a2 - 9d2 ) ( a2 - d2 ) = 384
⇒ a4 - a2d2 - 9a2d2 + 9d4 = 384
⇒ a4 - 10a2d2 + 9d4 = 384 , Now we substitute a = 5 ,and get
⇒ 54 - 10 × 52 × d2 + 9d4 = 384
⇒ 625- 250d2 + 9d4 = 384
⇒ 9d4 - 250d2 + 241 = 0
from , Splitting the middle term method we get
⇒ 9d4 - 9d2 - 241d2 + 241 = 0
⇒ 9d2 ( d2 - 1 ) - 241 ( d2 - 1 ) = 0
⇒ ( 9d2 - 241 ) ( d2 - 1 ) = 0
So,
d =± 1
And
d =± 241√3
But as our numbers are natural numbers , So we take
d = 1 or - 1 to get natural numbers of our A.P.
Our four terms of A.P. are
⇒ 5 - 3× 1 = 2
⇒ 5 - 1 = 4
⇒ 5 + 1 = 6
and
⇒ 5 + 3 × 1 = 8
SO,
Four terms of A.P. that gives, their sum is 20 and product is 384 = 2 , 4 , 6 , 8
Kindly find the solution to your query:
Let First term of A.P. = a
And
Common difference = 2d
So
Four numbers in A.P. = ( a - 3d ) , ( a - d ) , ( a + d ) and ( a + 3d )
So, given
And
from , Splitting the middle term method we get
So,
d =
And
d =
But as our numbers are natural numbers , So we take
d = 1 or - 1 to get natural numbers of our A.P.
Our four terms of A.P. are
and
SO,
Four terms of A.P. that gives, their sum is 20 and product is 384 = 2 , 4 , 6 , 8
To practice with some more questions following are the links of the similar queries as asked by you:
https://www.meritnation.com/ask-answer/question/three-positive-integers-a1-a2-a3-are-in-ap-such-that-a1/arithmetic-progressions/1502140
https://www.meritnation.com/ask-answer/question/q1-find-the-3-members-of-an-ap-whose-sum-is-20-and-product/math/3543750
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