Sum of four numbers in an AP is 86 and sum of their squares is 1894. Find the numbers
Answers
Step-by-step explanation:
let the four numbers be
x, x+3,x+6,x+9,
Sum of four numbers in an AP is 86
x+x+3+x+6+x+9=86
4x+18=86
4x=68
x=17
the numbers are
17,20,23,26.
and sum of the square of the numbers
289+400+529+676=1894
Let the four numbers be,
a - 3d, a - d, a + d, a + 3d
( The common difference is 2d)
Given,
Sum of the four numbers is 86.
(a - 3d) + (a - d) +(a + d) + (a + 3d) = 86
4a = 86
2a = 43
We shall use this later on again.
Now, Given
Sum of their squares is 1894
We have,
(a - 3d)² + (a - d)² +(a + d)² + (a + 3d)² = 1894
On simplification we get ;
4a² + 20d² = 1894
(2a)² + 20d² = 1894
20d² = 1894 - (43)²
20d² = 1894 - 1849 = 45
d² = 9/4
d = ± 3/2
So 2d = ±3
Now our numbers are, ( Taking 2d = +3 )
a - 3d = 2( a - 3d)/2 = (2a - 6d)/2 = 43 - 9/2 = 34/2 = 17
Next numbers are 17 + 3, 17 + 6, 17 + 9
The set of numbers would thus be, 17, 20, 23, 26
If we consider 2d = - 3 then,
a - 3d = 2 ( a - 3d)/2 = 43 + 9 / 2 = 52/2 = 26.
The other numbers would be 26 - 3, 26 - 6, 26-9
The set of numbers would thus be 26, 23, 20, 17.