A student states that for any positive integer n the sum of n successive integers is always divisible by n. Which of the following can be used to disprove the student's statements?
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)
I'm torn between A and E and not sure really how to solve.
In order to disprove the student's statement, we need to find a counterexample - a case where the sum of n successive integers is not divisible by n.
Let's analyze each option:
A) 1+2=3: This is a sum of 2 successive integers. The sum is 3, and since 3 is not divisible by 2, this option does disprove the student's statement.
B) 1+2+3=2x3: This is a sum of 3 successive integers. The sum is 6, and since 6 is divisible by 3, this option does not disprove the student's statement.
C) 1+2+3+4+5=3x5: This is a sum of 5 successive integers. The sum is 15, and since 15 is divisible by 5, this option does not disprove the student's statement.
D) p+(p+1)+(p+2)=3(p+1): This is a sum of 3 successive integers where p is a variable representing any positive integer. Simplifying this expression, we get 3p + 3, which is a multiple of 3. Thus, this option does not disprove the student's statement.
E) (p+1)+(p+2)+(p+3)+(p+4)+(p+5)=5(p+3): This is a sum of 5 successive integers where p is a variable representing any positive integer. Simplifying this expression, we get 5p + 15, which is always divisible by 5. Thus, this option does not disprove the student's statement.
From the analysis above, we see that option A is the correct answer as it provides a counterexample where the sum of 2 successive integers is not divisible by 2.