Four numbers are in A.P. The sum of first and last
is 8 and the product of both middle terms is 15.
The least among the four numbers is
(2) 2
(1) 1
(4) 4
Answers
Four numbers are in AP. The sum of first and last term is 8 and the product of middle terms is 15. The lest numbers of the series is???
Report by Mmm122 13.05.2018
Answers
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Let first number is 'a' and common difference is 'd'.
A.P. = a , a + d , a + 2d , a + 3d.
Given,
First term + Last term = 8
a + a + 3 d = 8
2 a + 3 d = 8
2 a = 8 - 3d -------- eq.
a = ( 8 - 3 d ) / 2
Now,
Product of middle terms = 15
( a + d ) ( a + 2 d ) = 15
a ( a + 2 d ) + d ( a + 2 d ) = 15
a^2 + 2 ad + ad + 2 d^2 = 15
a^2 + 3 ad + 2 d^2 = 15.
By substituting the value of 'a' in above eq.
{ ( 8 - 3 d ) / 2 }^2 + 3 { ( 8 - 3 d ) / 2 } d + 2 d^2 = 15
64 + 9 d^2 - 48 d 24 d - 9 d^2
-------------------------- + --------------------- + 2 d^2 = 15
4 2
64 + 9 d^2 - 48 d + 48 d - 18 d^2 + 8 d^2
--------------------------------------------------------- = 15
4
64 - d^2
------------------- = 15
4
64 - d^2 = 15 x 4
64 - d^2 = 60
64 - 60 = d^2
d^2 = 4
d = √4
d = ±2.
So, d = 2 or - 2.
By putting the value of ( d = 2 ) in eq.1
2 a + 3 d = 8
2 a + 3 ( 2 ) = 8
2 a + 6 = 8
2 a = 8 - 6
2 a = 2
a = 2/2 = 1.
If the common difference of a term is in ( + ) , then first term will be the least number of the series .
So, the least number of this series is 1.
By substituting the value of ( d = -2 ) in eq.1
2 a + 3 d = 8
2 a + 3 ( -2 ) = 8
2 a - 6 = 8
2 a = 8 + 6
2 a = 14
a = 14 / 2 = 7.
Last term = a + 3 d = 7 + 3 ( - 2 ) = 7 - 6 = 1.
If the common difference of an A.P is in ( - ) , then lat term will be the least number .
In all cases we found that least number is 1.
The least number of this series is 1.