A student states that for any positive integer n is the sum of n successive integers is always divisible by n. Which of the following can be used to disprove the student's statement?
Please provide answer and how you got it!
A) 1+2=3
B) 1+2+3=2x3
C) 1+2+3+4+5=3x5
D) p+(p+1)+(p+2)=3(p+1)
E) (p+1)+(p+2)+(p+3)+(p+4)+(p+5)=5(p+3)
how about (A)? 3 is not divisible by 2.
the answer is a
To disprove the student's statement, we need to find a counterexample where the sum of n successive integers is not divisible by n.
Let's consider each option:
A) 1 + 2 = 3
The sum of 1 and 2 is 3, which is not divisible by 2. However, this counterexample is not relevant since n = 2 in this case, and the statement claims that it holds true for any positive integer n.
B) 1 + 2 + 3 = 2x3
The sum of 1, 2, and 3 is 6, which is divisible by 3. This does not disprove the student's statement, as it supports it.
C) 1 + 2 + 3 + 4 + 5 = 3x5
The sum of 1, 2, 3, 4, and 5 is 15, which is divisible by 5. This does not disprove the student's statement, as it supports it.
D) p + (p+1) + (p+2) = 3(p+1)
This option involves a variable p, so we cannot directly evaluate it. However, we can substitute a value for p and see if the equation holds for that value. Let's take p = 1:
1 + (1+1) + (1+2) = 3(1+1)
1 + 2 + 3 = 3(2)
6 = 6
In this case, the equation holds for p = 1. However, this does not disprove the student's statement, as it only provides an example where the equation holds true.
E) (p+1) + (p+2) + (p+3) + (p+4) + (p+5) = 5(p+3)
Similar to option D, this option involves a variable p. Substituting a value for p, let's take p = 1:
(1+1) + (1+2) + (1+3) + (1+4) + (1+5) = 5(1+3)
2 + 3 + 4 + 5 + 6 = 5(4)
20 = 20
Again, the equation holds for p = 1, but it does not disprove the student's statement.
After evaluating all options, we can see that none of them disprove the student's statement. Therefore, the correct response is that none of the given options can be used to disprove the student's statement.