Formula Forensics 022. Sum the Odd Numbers between 1 and 100

Posted on May 24th, 2012 in Formula Forensics , Huis , Posts by Hui - 26 comments

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:

  1. First make an array of all numbers from 1 to 50
  2. Second double the array values
  3. Subtract 1.
  4. 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 Forensic Series

 

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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26 Responses to “Formula Forensics 022. Sum the Odd Numbers between 1 and 100”

  1. 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)

  2. Chiquitin says:

    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

  3. Thyson says:

    My suggestion:

    =SUM(IF(MOD(A2:A10,2)=0,A2:A10))  Ctrl Shift Enter

  4. Pedro Wave says:

    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))
     

  5. Vishwa says:

    This is replacing offset value
    =SUM(2*ROW(1:50)-1)

  6. You may as well  avoid a control.shift.enter using sumProduct().

    =SUMPRODUCT(ROW(1:100)*( ISODD(ROW(1:100) )))
     
    Bruce 

  7. Matthew Holbrook says:

    =SUMPRODUCT(ROW(1:100)*ISODD(ROW(1:100)))

  8. 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 

  9. Eric says:

    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.

  10. Eddie says:

    Sum the uneven numbers :
     
    =SUM(ROW(1:100)*ISODD(ROW(1:100))) ?
     
    Or am I missing something ?
    (just an amateur :))

  11. Jason H says:

    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.     

  12. modeste says:

    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))

  13. Works for all positive integers

      =(2*ROUND(100/2,0))^2/4

  14. Sudhir Gawade says:

    look at this without CSE
    A1=N
    =(CEILING(A1,2)/2)^2

  15. rags6 says:

    You are very creative at your side, a pretty nice way of computing values. Good job!
    http://www.techtipsntricks.com/

  16. %MakSo% says:

    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.   

  17. Malcolm says:

    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! 

  18. Omalohoso says:

    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.

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