Summing Consecutive Numbers

Consecutive numbers, consecutive odd numbers, consecutive even numbers. The finite sum of the consecutive numbers. The formulas of the finite sums of the consecutive numbers. The subject expression and solved questions.



Consecutive Numbers

Consecutive numbers are numbers that increase according to a certain rule. This amount of increase may be in the form 1, 2, 3, 9, …, etc.


If consecutive numbers increase one by one and cannot be divided exactly by 2, this numbers are said "consecutive odd numbers".

If consecutive numbers increase by twos and can be divided exactly by 2, this numbers are said "consecutive even numbers".

 


Consecutive Numbers That Increasing One By One

n is an integer,

1, 2, 3, 4, 5, 6, … , n

Numbers are consecutive counting numbers.

Also this numbers are consecutive positive integers.


The formula of the sum of numbers from 1 to n.

T =n.(n + 1)
2




Example:

Find the sum of numbers from 1 to 35.


Solution:

The formula of the sum of numbers from 1 to n.

T = n•(n+1)
2




For n = 35,

T = 35•36
2




T = 630


Example:

Find the sum of consecutive numbers from 30 to 60 (inclusive of 30 and 60)


Solution:

Firstly, let we find the sum of numbers from 1 to 29. Later, let we find the sum of numbers from 1 to 60.

The difference between these two numbers will give us the sum of numbers from 30 to 60.

T1 = 29•30
2




T1 = 435

T2 = 60•61
2




T2 = 1830

T = 1830 – 435

T = 1395


Consecutive Odd Numbers


n is an integer,

1,3, 5, 7, 9, 11, 13,15, 17, ... (2n-1)


Numbers are called "consecutive odd numbers".


The formula for the sum of consecutive odd numbers.

T = n•n 

T = n2

Finding n,

n = Last term - first term + 1
2




Example:

Find the sum of consecutive odd numbers from 1 to 15.


Solution:

The formula is 

T = n2

n = Last term - first term + 1
2




n = 15 - 1 + 1
2



n = 8

Sum = 82

Sum = 64


Example:

Find the sum of consecutive odd numbers from 19 to 75.


Solution:

Firstly, we find the sum of consecutive odd numbers drom 1 to 17. Later, we find the sum of consecutive odd numbers 1 to 75. he difference of these two sum gives us the sum of consecutive odd numbers from 19 to 75.


n = 17 - 1 + 1
2




n = 9

T1 = 92

T1 = 81

m = 75 - 1 + 1
2




m = 38

T2 = 382

T2 = 1444


T = T2 – T1

T = 1444 – 81

T = 1363


Consecutive Even Numbers

If consecutive numbers increase by twos and can be divided exactly by 2, this numbers are said "consecutive even numbers".

Consecutive even numbers are in the form of 

2, 4, 6, 8, 10, 12, 14, 16, 18, 20, … 2n


The sum of consecutive even numbers is founds by equality below.

Total = n•(n + 1)

Finding n,

n = Last term - first term + 1
2




Example:

Find the sum of consecutive even numbers from 2 to 90.


Solution:

Finding the n,

2n = 90

n = 45

Or

n = Last term - First term + 1
2




Last term = 90

First term = 2

n = 90 - 2 + 1
2




n = 45


Total = 45 • 46

= 2070



The General Formula Of the Sum Of The Consecutive Numbers


The consecutive numbers may not always be as regular as above explanations.

Now, we will give a general formula that can used for the sum of the all consecutive numbers.


T = The sum of the consecutive numbers


Consecutive_S1I1



The Number Of Terms

r + (x•1 + r) + (x•2 + r) + … + (x•n + r) in the number array,


m = Last term

m = x•n + r


Nt = Number of terms,

Consecutive_S1I2


Example:

What is the sum of the numbers that starting from 1 and continuing up to 64 and increasing three by three?

Solution:


The first term = 1

The last term = 64

The amount increase = 3

Nt = Number of terms,

Consecutive_S1I2b


Nt = 64 - 1 + 1
3



Nt = 22



T = The sum of the consecutive numbers

Consecutive_S1I1b


T =22•65
2



T = 11•65

T = 715


Example:

What is the sum of the numbers that starting from 12 and continuing up to 117 and increasing seven by seven.


Solution:


The first term = 12

The last term = 117

The amount of increase = 7


Nt = The number of terms

Nt = 117 – 12 + 1
7




Nt = 15 + 1

Nt = 16

 

Sum =16 • (117 + 12)
2




Total = 1032


Example:


The sum of three consecutive integers is 72. The amount of increase of these numbers is 4.

What is the smallest these numbers.


Solution:

Suppose that smallest of these numbers is x.

In this case,

Second number is (x + 4)

Third number is (x + 8)

Sum = 72

72 = x + (x + 4) + (x + 8)

72 = 3x + 12

60 = 3x

x = 20





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17/07/2018

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