Question

find four numbers in AP whose sum is 20 and the sum of whose square is 120

Answer 1

numbers are 2,4,6,8
2+4+6+8==20

2^2 + 4^2 + 6^2 +8^2 == 120

Answer 2

Heya !!!


Let the required Numbers be ( A - 3D ) , ( A - D ) , ( A + D ) and ( A + 3D ).



Then,



A - 3D + A - D + A + D + A + 3D = 20



4A= 20


A = 20/4 = 5


And,



( A - 3D)² + ( A - D )² + ( A + D )² + ( A + 3D )² = 120



=> 4 ( A² + 5D² ) = 120



=> ( A² + 5D² ) = 120/4



=> ( A² + 5D² ) = 30


=> (5)² + 5D² = 30


=> 5D² = 30-25


=> 5D² = 5


=> D² = 1


=> D = ✓1 = 1 or -1


Thus,

A = 5 and D = (+-) 1


Hence,



First number = A - 3D = 5 - 3 × 1 = 2


Second number = A - D = 5 - 1 = 4


Third Number = A + D = 5+1 = 6


And,


Fourth Number = A + 3D = 5 + 3 × 1 = 8.

The required Numbers are ( 2 , 4 , 6 , 8 ) or ( 8 , 6 , 4 , 2).



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