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Formula Forensics 022. Sum the Odd Numbers between 1 and 100
Last week at the Chandoo.org Forums, Sunita, posed the question:
“Please help me to find out the sum of odd numbers in a range of 1-100 numbers
Like 1+3+5+7+ … 97+ 99
How it will find through an excel formula?”
I chipped in with two array formulas:
=SUM(2*ROW(OFFSET($A$1,,,100/2))-1) Ctrl Shift Enter
and
=SUM(ROW(1:100)*MOD(ROW(1:100),2)) Ctrl Shift Enter
Lets look at each of these in turn.
As usual at Formula Forensics you can download a Sample File here and follow along Download Sample File.
Formula 1: =SUM(2*ROW(OFFSET($A$1,,,100/2))-1)
The first formula we will examine is:
=SUM(2*ROW(OFFSET($A$1,,,100/2))-1) Ctrl Shift Enter
This formula works on the principle of making an array of the odd numbers between 1 and 100 and the adding them up.
We can make an array of the odd numbers from 1 to 100 by:
- First make an array of all numbers from 1 to 50
- Second double the array values
- Subtract 1.
- Add up the values
1. Make an Array from 1 to 50
The formula ROW(OFFSET($A$1,,,100/2)) can be used to make an array of the numbers from 1 to 50
In a spare cell, D4, type =ROW(OFFSET($A$1,,,100/2)) then press F9 not enter
Excel will respond with an array: ={1;2;3;4;5;6;7;8;9;10;11;12;13;14;15;16;17;18;19;20;21;22;23;24;25;26;27;28;29;30;31;32;33;34;35;36;37;38;39;40;41;42;43;44;45;46;47;48;49;50}
How does this work?
Offset($A$1,,,100/2) sets up a Range from A1 with no Row or Column offset, but with a height of 100/2 = 50.
In a spare cell, D6, type =OFFSET($A$1,,,100/2) then press F9 not enter
Excel will respond with an array: ={0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0}
We can see that the array contains 50 Zero’s (You can count them to check).
True.
But it is 50 Rows of Zero’s. The ;’s in the array separate Rows.
So the expanded formula: =ROW(OFFSET($A$1,,,100/2))
Returns the Rows of the Array Elements, not the Array Values.
2. Double the Values
The formula 2*ROW(OFFSET($A$1,,,100/2)) is used to double the array values
In a spare cell, D8, type =2*ROW(OFFSET($A$1,,,100/2)) then press F9 not enter
Excel will respond with an array: ={2;4;6;8;10;12;14;16;18;20;22;24;26;28;30;32;34;36;38;40;42;44;46;48;50;52;54;56;58;60;62;64;66;68;70;72;74;76;78;80;82;84;86;88;90;92;94;96;98;100}
3. Subtract 1
The formula 2*ROW(OFFSET($A$1,,,100/2)) -1 is used to subtract a value of a from the array values
In a spare cell, D10, type =2*ROW(OFFSET($A$1,,,100/2)) -1 then press F9 not enter
Excel will respond with an array: ={1;3;5;7;9;11;13;15;17;19;21;23;25;27;29;31;33;35;37;39;41;43;45;47;49;51;53;55;57;59;61;63;65;67;69;71;73;75;77;79;81;83;85;87;89;91;93;95;97;99}
4. Add up the Values
The formula =Sum(2*ROW(OFFSET($A$1,,,100/2)) -1) is used to add up the array values
In a spare cell, D12, type =Sum(2*ROW(OFFSET($A$1,,,100/2)) -1) then press F9 not enter
Excel will respond with a value of = 2500, The Answer.
Formula 2: =SUM(ROW(1:100)*MOD(ROW(1:100),2))
The second formula we will examine is:
=SUM(ROW(1:100)*MOD(ROW(1:100),2)) Ctrl Shift Enter
This formula works by constructing an array of values between 1 and 100 and then multiplying that Array by an Array of the Odd values between 1 and 100 and then adding up the resultant Array.
Lets start with: =SUM(ROW(1:100)*MOD(ROW(1:100),2))
Note that the Row(1:100) is used twice in the formula.
In a spare cell: D17 type: =Row(1:100) then press F9 not enter
Excel will respond with an array: ={1;2;3;4;5;6;7;8;9;10;11;12;13;14;15;16;17;18;19;20;21;22;23;24;25;26;27;28;29;30;31;32;33;34;35;36;37;38;39;40;41;42;43;44;45;46;47;48;49;50;51;52;53;54;55;56;57;58;59;60;61;62;63;64;65;66;67;68;69;70;71;72;73;74;75;76;77;78;79;80;81;82;83;84;85;86;87;88;89;90;91;92;93;94;95;96;97;98;99;100}
An array of the values from 1 to 100.
Next we will look at the Mod section =SUM(ROW(1:100)*MOD(ROW(1:100),2))
In a spare cell: D19 type: =Mod(Row(1:100),2) then press F9 not enter
Excel will respond with an array: ={1;0;1;0;1;0;1;0;1;0;1;0;1;0;1;0;1;0;1;0;1;0;1;0;1;0;1;0;1;0;1;0;1;0;1;0;1;0;1;0;1;0;1;0;1;0;1;0;1;0;1;0;1;0;1;0;1;0;1;0;1;0;1;0;1;0;1;0;1;0;1;0;1;0;1;0;1;0;1;0;1;0;1;0;1;0;1;0;1;0;1;0;1;0;1;0;1;0;1;0}
Mod returns the remainder after dividing the first parameter by the second
Eg: Mod(5,2)=1 5 divided by 2 = 2 Remainder 1.
So in our example Mod( Array, 2 ) returns the value 1 for the Odd Values and 0 for the Even values.
Next we multiply the 2 arrays together: =SUM(ROW(1:100)*MOD(ROW(1:100),2))
This is the same as:
={1;2;3;4;5; … ;97;98;99;100} * {1;0;1;0;1; … ;1;0;1;0}
In a spare cell: D21 type: =ROW(1:100)*MOD(ROW(1:100),2) then press F9 not enter
Excel will respond with an array: ={1;0;3;0;5;0;7;0;9;0;11;0;13;0;15;0;17;0;19;0;21;0;23;0;25;0;27;0;29;0;31;0;33;0;35;0;37;0;39;0;41;0;43;0;45;0;47;0;49;0;51;0;53;0;55;0;57;0;59;0;61;0;63;0;65;0;67;0;69;0;71;0;73;0;75;0;77;0;79;0;81;0;83;0;85;0;87;0;89;0;91;0;93;0;95;0;97;0;99;0}
Finally we can add up the array values: =SUM(ROW(1:100)*MOD(ROW(1:100),2))
In a spare cell: D23 type: =SUM(ROW(1:100)*MOD(ROW(1:100),2)) then press F9 not enter
Excel will respond with a value of = 2500, The Answer.
Variation 1:
In the above formula =SUM(ROW(1:100)*MOD(ROW(1:100),2)) we described a method of evaluating Array values as either Odd or Even using the Mod function.
Excel has a built in function for determining if a Value is Odd and that is Isodd()
We can modify the above equation to use Isodd() as follows
=SUM(ROW(1:100)*ISODD(ROW(1:100)))
You can check it in cell D28.
What if I want to Sum the Even numbers?
We can use the variation described above to quickly add up the even numbers between 1 and 100
=SUM(ROW(1:100)*ISEVEN(ROW(1:100)))
In a spare cell: D21 type: =SUM(ROW(1:100)*ISEVEN(ROW(1:100))) then press F9 not enter
Excel will respond with a value of = 2550, The Answer.
How Else Can You Solve Sunita’s Problem?
Can you solve Sunita’s problem another way?
Let us know in the comments below:
Download
You can download a copy of the above file and follow along, Download Here.
Formula Forensics “The Series”
This is the 22nd post in the Formula Forensics series.
You can learn more about how to pull Excel Formulas apart in the following posts
Formula Forensics Needs Your Help
I need more ideas for future Formula Forensics posts and so I need your help.
If you have a neat formula that you would like to share and explain, try putting pen to paper and draft up a Post like above or;
If you have a formula that you would like explained, but don’t want to write a post, send it to Hui or Chandoo.
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At Chandoo.org, I have one goal, "to make you awesome in Excel & Power BI". I started this website in 2007 and today it has 1,000+ articles and tutorials on data analysis, visualization, reporting and automation using Excel and Power BI. 
26 Responses to “Formula Forensics 022. Sum the Odd Numbers between 1 and 100”
For those who might be interested, there is a direct formula (no arrays) to calculate the first N odd numbers. Assuming A1 contains the number (N), this formula will perform the desired calculation...
=A1*(A1+1)/2-INT(A1/2)*(INT(A1/2)+1)
Why not use the mathematical formulas that already exist?
For an arithmetic progression, the sum of the n first terms (S) is:
S = n* (a1+an)/2
a1 is the first term of the progression and an is the nth term.
S = 50*(1+99)/2 for odd numbers
S = 50*(2+100)/2 for even numbers
My suggestion:
=SUM(IF(MOD(A2:A10,2)=0,A2:A10)) Ctrl Shift Enter
Assuming A1 contains the number (N)
Sum the first N numbers:
=A1*(1+A1)/2
Sum the first odd numbers:
=POWER(TRUNC((A1+1)/2),2)
Sum the first even numbers:
=POWER(TRUNC(A1/2),2)+TRUNC(A1/2)
=TRUNC(A1/2)*(1+TRUNC(A1/2))
This is replacing offset value
=SUM(2*ROW(1:50)-1)
Vishwa, your formula assuming A1 contains the number (N)
=SUM(2*ROW(INDIRECT("1:"&TRUNC((A1+1)/2)))-1) Ctrl Shift Enter
You may as well avoid a control.shift.enter using sumProduct().
=SUMPRODUCT(ROW(1:100)*( ISODD(ROW(1:100) )))
Bruce
=SUMPRODUCT(ROW(1:100)*ISODD(ROW(1:100)))
There's no need to use array formulas:
=SUMPRODUCT(ROW(1:100)*ISODD(ROW(1:100)))
Another way would be:
=ROUNDUP(100/2,0)^2
The sum of the first n odd numbers is n squared (i.e., n^2).
n = 1 => sum = 1 =1^2
n = 2 => 1 + 3 = 4 =2^2
n = 3 => 1 + 3 + 5 = 9 =3^2
etc
The sum of the first n even (and positive) numbers is n * (n+1)
n = 1 => sum = 2 = 1 * (1 + 1) = 1 * 2 = 2
n = 2 => 2 + 4 = 6 = 2 * (2 + 1) = 2 * 3 = 6
n = 3 => 2 + 4 + 6 = 12 = 3 * (3 + 1) = 3 * 4 = 12
etc
Not really an issue for using Excel... but an issue for using math skills.
Sum the uneven numbers :
=SUM(ROW(1:100)*ISODD(ROW(1:100))) ?
Or am I missing something ?
(just an amateur :))
In the same vein as others who've pointed out this is a mathematical progression one way of producing the same result would be the following formula:
=ROUNDUP(n/2,0)^2
This version should work for all cases of determining sum of odd number whether provided with an even or odd number to target (n).
I remember at schoold learning the the formula for the number triangle, i.e. 1+2+3+4+...n
which is:
=(n^2 + n)/2
and this has a common basis, but I'm not a mathemetician so I could be wrong.
Regards JH
> I remember at schoold learning the the formula for the number triangle,
> i.e. 1+2+3+4+…n
>
> which is:
>
> =(n^2 + n)/2
I have always found this easier to remember when written like this...
=n*(n+1)/2
written this way, it is easy to remember we are multiplying the number n by the number that follows it (n+1) and then dividing that product by 2. Since either the number or the number following it must be even, we see the division by 2 works with no remainder. Also thinking about this, you can almost do the calculation in your head. Sum of the first 20 numbers.... 20*21/2... but performing 20/2 (yielding 10) is easy to do in your head and multiplying 21 by that resulting 10 is a snap... answer almost immediately is 210.
Hi folks...
may be i'm nut
but i understand to compute cells values either row number ???
the sum of Oddvalue in a continues number serie is a mathematical function as Rick says.
to compute Odd cells values i use :
=SUMPRODUCT(Myrange*(MOD(MyRange,2)=0))
Works for all positive integers
=(2*ROUND(100/2,0))^2/4
look at this without CSE
A1=N
=(CEILING(A1,2)/2)^2
=ROUND(F13/2,0)^2
Or even shorter: (EVEN(F13)/2)^2
This must be the shortest way to do it in Excel ...?
JeanBar ... Thats Brilliant
JeanBar the Even is not required.
If A1 contains 100 then (A1/2)^2 works as well
Sam,
The EVEN() function is necessary. Your formula works for 100 indeed but not with 99: the result would be 2450.25. You need to have an integer result whatsoever.
You are very creative at your side, a pretty nice way of computing values. Good job!
http://www.techtipsntricks.com/
If C28 is the cell where I have a number up to which I'll add up the odd natural numbers - I will use the following:
"=IF(ISEVEN(C28),C28*(C28+1)/2-(C28/2)*(C28/2+1),C28*(C28+1)/2-((C28-1)/2)*((C28-1)/2+1))"
using the principle sum of first n natural numbers = n*(n+1)/2
when the value is even - such as 100, I sum up all natural numbers to 100 using C28*(C28+1)/2. Then, see the even numbers 2,4,...100 are nothing but double the sum of first 50 natural numbers. Thus I subtract 2* (C28/2)*(C28/2+1)/2.
If the given number is odd, I follow the same way, except I figure out C28 - 1 to be the number up to which I'll sum the even natural number series.
Simplest formula:
= 25 * 100
Surely this is a case where the cleverest solution is to think about it first and see you don't need a formula? Adding up the numbers 1,3,5,7....97,99 is easiest if you think of it as (1+99),(3+97),(5+95).... there are clearly 25 pairs of 100, giving the answer 2500 in the time it takes to just step back and think.
Sometimes the simplest solution is the most elegant!
May the Grace and peace from above abide with you all as you are trying to input some helpful notes in order to help other people around the World.