c210 wrote:
Hey guys and gals,
question: Is X^2 odd?
statement 2: The sum of x consecutive integers is divisible by x
the solution states that the sum divided by x is an integer and moreover the sum of x consecutive integers divided by x is the average of that set of x integers. this therefore proves that x^2 is odd....how does that make sense?
anyone care to explain?
C210
Hi there,
When posting a question in PS/DS subforums make sure that the question is complete. So, please edit the question you've posted and add statement (1).As for your query, below are some tips about consecutive integers (for more on this topic see Number Theory Chapter of Math Book:
math-number-theory-88376.html).
Consecutive IntegersConsecutive integers are integers that follow one another, without skipping any integers. 7, 8, 9, and -2, -1, 0, 1, are consecutive integers.
• Sum of \(n\) consecutive integers equals the mean multiplied by the number of terms, \(n\). Given consecutive integers \(\{-3, -2, -1, 0, 1,2\}\), \(mean=\frac{-3+2}{2}=-\frac{1}{2}\), (mean equals to the average of the first and last terms), so the sum equals to \(-\frac{1}{2}*6=-3\).
• If n is odd, the sum of consecutive integers is always divisible by n. Given \(\{9,10,11\}\), we have \(n=3\) consecutive integers. The sum of 9+10+11=30, therefore, is divisible by 3.
• If n is even, the sum of consecutive integers is never divisible by n. Given \(\{9,10,11,12\}\), we have \(n=4\) consecutive integers. The sum of 9+10+11+12=42, therefore, is not divisible by 4.
• The product of \(n\) consecutive integers is always divisible by \(n!\).
Given \(n=4\) consecutive integers: \(\{3,4,5,6\}\). The product of 3*4*5*6 is 360, which is divisible by 4!=24.
So, if statement (2) says that "the sum of x consecutive integers is divisible by x", according to the second point above x is odd, which makes x^2 an odd number.
As for your second question:
A set of consecutive integers is an evenly spaced set (AP).
• In any evenly spaced set the arithmetic
mean (average) is equal to the median.
• The sum of the elements in any evenly spaced set equals to the mean multiplied by the number of terms: {sum}={mean}*{# of terms}
Thus, for original question: sum/x must equal to the mean (and median) and as median of odd consecutive integers is an integer itself, then sum/x=mean=median=integer.
Hope it's clear.
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