Formulas for Consecutive Integers, Even integers, Odd Integers, etc
Consecutive Integers
DEFINITION:
Consecutive integers are integers that follow one another, without skipping any integers. So, "consecutive integers" ALWAYS mean integers that follow each other in order with common difference of 1: ... x-3, x-2, x-1, x, x+1, x+2, ....
Examples:
7, 8, 9, and -2, -1, 0, 1, are consecutive integers.
2, 4, 6 ARE NOT consecutive integers, they are consecutive even integers.
3, 5, 7 ARE NOT consecutive integers, they are consecutive odd integers.
FORMULAS:
• Sum of \(n\) consecutive integers equals the mean multiplied by the number of terms, \(n\): \(Sum=mean*number \ of \ terms\).
Example: 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 \(-\frac{1}{2}*6=-3\).
PROPERTIES:
• If n is odd, the sum of consecutive integers is always divisible by n.
Example:: 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.
Example: 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 \(n!\).
Example: given \(n=4\) consecutive integers: \(\{3,4,5,6\}\). The product of 3*4*5*6 is 360, which is divisible by 4!=24.
Evenly Spaced Set
DEFINITION:
Evenly spaced set or an arithmetic progression is a sequence of numbers such that the difference of any two successive members of the sequence is a constant.
Examples: the set of integers \(\{9,13,17,21\}\) is an example of evenly spaced set with common difference of 4. Set of consecutive integers, {-4, -2, 0, 2, 4, 6, ...} is also an example of evenly spaced set with common difference of 2.
FORMULAS:
• If the first term is \(a_1\) and the common difference of successive members is \(d\), then the \(n_{th}\) term of the sequence is given by:
\(a_ n=a_1+d(n-1)\)
• In any evenly spaced set the arithmetic mean (average) is equal to the median and can be calculated by the formula \(mean=median=\frac{a_1+a_n}{2}\), where \(a_1\) is the first term and \(a_n\) is the last term.
Examples: given the set \(\{7,11,15,19\}\), \(mean=median=\frac{7+19}{2}=13\).
• The sum of the elements in any evenly spaced set is given by: \(Sum=\frac{a_1+a_n}{2}*n\), the mean multiplied by the number of terms. OR, \(Sum=\frac{2a_1+d(n-1)}{2}*n\)
• If the evenly spaced set contains odd number of elements, the mean is the middle term, so the sum is middle term multiplied by number of terms.
Example: There are five terms in the set {1, 7, 13, 19, 25}, middle term is 13, so the sum is 13*5 =65.
Formulas For Special Cases:
Sum of n first positive integers: \(1+2+...+n=\frac{1+n}{2}*n\)
Example: given \(n=4\) the sum of four first positive integers \(1+2+3+4=\frac{1+4}{2}*4=10\).
Sum of n first positive ODD numbers: \(a_1+a_2+...+a_n=1+3+...+a_n=n^2\), where \(a_n\) is the last, \(n_{th}\) term and given by: \(a_n=2n-1\).
Example: given \(n=5\) first odd positive integers, then their sum equals to \(1+3+5+7+9=5^2=25\).
Sum of n first positive EVEN numbers: \(a_1+a_2+...+a_n=2+4+...+a_n\)\(=n(n+1)\), where \(a_n\) is the last, \(n_{th}\) term and given by: \(a_n=2n\).
Example: given \(n=4\) first positive even integers, then their sum equals to \(2+4+6+8=4(4+1)=20\).
MORE THEORY AND QUESTIONS:
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