three number are in gp whose sum is 70.if the extremes are multiplied by 4 and the middle by 5,the result is in AP .find the nos.
Answers
EXPLANATION.
- GIVEN
Three number Are in Gp whose sum = 70.
the extremes are multiply by 4 and the middle
by 5 the results are in Ap
find the number.
According to the question,
Let the three number are in Gp.
=> a, ar, ar²
Case = 1
three number are in Gp whose sum is = 70
=> a + ar + ar² = 70
=> a ( 1 + r + r² ) = 70 ......(1)
case = 2
The extremes are multiply by 4 and the
middle by 5 results in Ap
=> 4a, 5ar, 4ar² ......Ap
conditions of an Ap.
=> 2b = a + c
=> 2 ( 5ar ) = 4a + 4ar²
=> 10ar = 4a + 4ar²
=> 4ar² - 10ar + 4a = 0
=> 2a ( 2r² - 5r + 2 ) = 0
=> 2r² - 5r + 2 = 0
=> 2r² - 4r - r + 2 = 0
=> 2r( r - 2 ) - 1 ( r - 2 ) = 0
=> ( 2r - 1 ) ( r - 2 ) = 0
=> r = 1/2 and r = 2
put the value of r in equation (1)
when r = 1/2
=> a ( 1 + 1/2 + (1/2)² ) = 70
=> a ( 1 + 1/2 + 1/4 ) = 70
=> a ( 4 + 2 + 1 / 4 ) = 70
=> a ( 7/4 ) = 70
=> a = 40
when r = 2
=> a ( 1 + 2 + (2)² ) = 70
=> a ( 1 + 2 + 4 ) = 70
=> a ( 7) = 70
=> a = 10
Therefore,
When a = 40 and r = 1/2
when a = 10 and r = 2
Therefore,
a = 40 and r = 1/2 number are =
=> 40 , 20 , 10
a = 10 and r = 2
=> 10 , 20 , 40
Answer:
Three number Are in Gp whose sum = 70.
the extremes are multiply by 4 and the middle
by 5 the results are in Ap
Step-by-step explanation:
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