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Math Expert V
Joined: 02 Sep 2009
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Formulas for Consecutive, Even, Odd Integers  [#permalink]

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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 $$2+4+6+8=4(4+1)=20$$.

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Joined: 23 Feb 2015
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Re: Formulas for Consecutive, Even, Odd Integers  [#permalink]

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Good initiative. Thanks bb.

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Location: United States
GRE 1: Q710 V430 WE: Information Technology (Computer Software)
Re: Formulas for Consecutive, Even, Odd Integers  [#permalink]

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Thanks bb for the inititaive.

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Re: Formulas for Consecutive, Even, Odd Integers  [#permalink]

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Sum of First "n" natural nos = n(n+1)/2
Sum of First "n" ODD natural nos = n^2
Sum of First "n" EVEN natural nos = n (n+1)
Sum of Square"n" natural nos = n(n+1)(2n+1)/6
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Resource: GMATPrep RCs With Solution Re: Formulas for Consecutive, Even, Odd Integers   [#permalink] 03 Feb 2019, 18:40
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