Find 4 numbers in ap whose sum is 20 and the sum of whose square is 120
In and why we take a-3d,a-d,a+d,a+3d
Answers
Solution :
Let a-3d,a-d,a+d,a+3d are
4 terms in A.P
According to the problem given,
a-3d+a-d+a+d+a+3d= 20
=> 4a = 20
=> a = 20/4
=> a = 5
(a-3d)²+(a-d)²+(a+d)²+(a+3d)²=120
=> 2[a² + (3d)²]+2[a²+d²] = 120
=> a² + 9d² + a² + d² = 60
=> 2a² + 10d² = 60
=> 2×5² + 10d² = 60
=> 50 + 10d² = 60
=> 10d² = 60 - 50
=> d² = 10/10
=> d² = 1
=> d = ± 1
Now ,
Required 4 numbers of A.P :
i ) if a = 5 , d = 1
(5-3),(5-1),(5+1),(5+3)
i.e , 2 ,4 , 6, 8
ii ) if a = 5 , d = -1
4 terms are
8,6,4,2
***********
We take a-3d,a-d,a+d,a+3d
to get first term ( a ) value
easily .
Otherwise it will be lengthy
procedure to solve the two
equations to get a , d values .
••••
Answer:
2 , 4 , 6 , 8
Step-by-step explanation:
Let say four numbers are
a - 3d , a -d , a + d , a + 2d
Reason why we take a-3d,a-d,a+d,a+3d
So that we sum all these numbers we get only on variable and we can find value of a
But it is not necessary to take number like this
we can take a , a+d , a+2d , a+3d also. check my alternate solution after its completion
Sum of a-3d,a-d,a+d,a+3d = 20
a -3d + a -d + a + d + a + 3d = 20
=> 4a = 20
=> a = 5
(a-3d)² + (a-d)² + (a+d)² + (a+3d)² = 120
=> a² + 9d² - 6ad + a² + d² - 2ad + a² + d² + 2ad + a² + 9d² + 6ad = 120
=> 4a² + 20d² = 120
=> a² + 5d² = 30
a = 5
=> 5² + 5d² = 30
=> 25 + 5d² = 30
=> 5d² = 30 -25
=> 5d² = 5
=> d² = 1
=> d = ±1
So numbers are
2 , 4 , 6 , 8 or 8 , 6 , 4 , 2
____________________________
here alternate solution
Let say 4 numbers are
a , a+d , a+2d , a + 3d
=> 4a + 6d = 20
=> 2a + 3d = 10 - eq 1
a² + (a+d)² + (a+2d)² + (a+3d)² = 120
=> a² + a² + d² + 2ad + a² + 4d² + 4ad + a² + 9d² + 6ad = 120
=> 4a² + 14d² + 12ad = 120 Eq2
Squaring Eq 1
(2a + 3d)² = 10²
=> 4a² + 9d² + 12ad = 100 Eq3
Eq 2 - Eq 3
=> 5d² = 20
=> d² =4
=> d = ±2
Now putting values of d in eq 1 , 2a + 3d = 10
Case 1 d = 2
2a + 6 = 10
=> 2a = 4
=> a = 2
numbers are 2 , 4 , 6 & 8
Case 2 d = -2
2a - 6 = 10
=> 2a = 16
=> a = 8
Numbers are 8 , 6 , 4 & 2