Find |S|: How to Calculate Sum of Integer Values of n

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In summary, the conversation discusses finding the sum of all integer values of n for which the expression (n^2+12n-43)/(n+6) is an integer. By completing the square in the numerator, it is found that the expression is an integer for only four values of n: -5, -7, 73, and -85. The absolute value of the sum, S, is 24.
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Homework Statement


Let S be the sum of all integer values of n such that [tex]\frac {n^2+12n-43} {n+6} [/tex] is an integer. What is the value of |S|.

Homework Equations


Since it's sum, S = [tex]\frac {n} {2} \ (2a+(n-1)d) [/tex] where a is the first term.
The [tex]\frac {n^2+12n-43} {n+6} [/tex] = x, where x is an integer

The Attempt at a Solution



[tex]\frac {n^2+12n-43} {n+6} [/tex]= x
[tex] {n^2+(12-x)n-43-6x}[/tex] = 0


basically I have no idea "Let S be the sum of all integer values of n" means?
Is it means that S = n or otherwise? Need clarification for this.
 

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  • #2
You need to find out for what values of ##n## the expression ##\frac {n^2+12n-43} {n+6} ## is an integer.

It's not an arithmetic progression.

Try defining ##m=n+6## and then express ##n^2+12n-43## in terms of ##m##.
 
  • #3
I would complete the square in the numerator. From that it turns out that the fraction is an integer for only a very small number of values of n!​
 
  • #4
I think I got it by completing the square.
it will becomes [tex]\frac {(n+6)^2 - 79}{n+6}[/tex]

Since 79 is prime number,
Solving n + 6 = ± 1 and n + 6 = ± 79 will gives rise to 4 integers, which is -5, -7, 73 and -85.
Hence, |S| = |-5-7+73-85| = 24

Thanks for pointing out the completing the square. It helps.
 

1. How do I calculate the sum of integer values of n?

To calculate the sum of integer values of n, you can use the formula: sum = (n * (n + 1)) / 2. Alternatively, you can also use a for loop to iterate through each integer value of n and add them together.

2. What is the purpose of finding the sum of integer values of n?

The sum of integer values of n is often used in mathematical and scientific calculations, such as in finding the average, median, or mode of a set of numbers. It can also help in analyzing patterns and relationships between different integer values.

3. Can negative values of n be included in the sum?

Yes, negative values of n can be included in the sum. The formula for calculating the sum of integer values of n can handle both positive and negative values.

4. Is there a limit to the number of integer values of n that can be included in the sum?

No, there is no limit to the number of integer values of n that can be included in the sum. However, if the sum becomes too large, it may exceed the maximum value that can be stored in a computer's memory.

5. Can I use this formula for any set of numbers, or only for consecutive integer values of n?

This formula is specifically for calculating the sum of consecutive integer values of n. If you want to find the sum of a different set of numbers, you will need to use a different formula or method.

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