The integer 90can be expressed as the sum of z consecutive integers. The value of z could be any of the following except
A) 3
B) 4
C) 5
D) 6
E) 9
If the first integer is n+1, the consecutive integers are: n+2, ...n+z.
If 90 is the sum of these z consecutive integers then...
90 = sum{i=1 to z} n+i
90 = zn + sum{i=1 to z} i
90 = zn + z(z+1)/2
.: n = (180-z(z+1))/2z
Testing the given values of z:
A) z=3 => n = (180-3*4)/6
B) z=4 => n = (180-4*5)/8
C) z=5 => n = (180-5*6)/10
D) z=6 => n = (180-6*7)/12
E) z=9 => n = (180-9*10)/18
One of these 'n' is not an integer.
One of these 'n' does not belong.
Can you tell me which one?
(D)6
6x + 5*6/2 = 90
6x + 15 = 90
6x = 75
x is not an integer
To determine the possible values of z, we can use the formula for the sum of an arithmetic series:
Sn = (n/2)(2a + (n-1)d)
where Sn is the sum of the series, n is the number of terms, a is the first term, and d is the common difference.
In this case, we are given that the sum is 90 and we want to find the possible values of z. Let's set up the equation:
90 = (z/2)(2a + (z-1))
Simplifying,
90 = (z/2)(2a + z - 1)
Since we want to express 90 as the sum of z consecutive integers, the first term (a) can be any integer. Since we are looking for possible values of z, we can't determine its exact value. However, we can determine the possible values for z by substituting different values for a.
To determine the possible values of z, we can check which values result in an integer solution for the equation.
For each possible value of a, we can solve the equation for z by substituting it into the equation:
90 = (z/2)(2a + z - 1)
Now, we can proceed by plugging in each answer choice for a and solving the equation to determine if it results in an integer solution for z.
A) If a = 3:
90 = (z/2)(6 + z - 1)
90 = (z/2)(z + 5)
This equation does not result in an integer solution for z.
B) If a = 4:
90 = (z/2)(8 + z - 1)
90 = (z/2)(z + 7)
This equation does not result in an integer solution for z.
C) If a = 5:
90 = (z/2)(10 + z - 1)
90 = (z/2)(z + 9)
This equation does result in an integer solution for z.
D) If a = 6:
90 = (z/2)(12 + z - 1)
90 = (z/2)(z + 11)
This equation does not result in an integer solution for z.
E) If a = 9:
90 = (z/2)(18 + z - 1)
90 = (z/2)(z + 17)
This equation does result in an integer solution for z.
Therefore, the possible values of z are:
A) 3 - not a possible solution
B) 4 - not a possible solution
C) 5 - a possible solution
D) 6 - not a possible solution
E) 9 - a possible solution
The answer is A) 3.