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Adding Sequential Numbers (III)

© Vidya Narayan Wadadekar

If you have added many even number sequences starting from 2 through a two digit even number you must have observed some pattern in your procedure.

What you are finally doing is, simply taking half of the two digit even number. Then you are multiplying this number with its immediate next number (which will always be an odd number). Thus, if you want to add all even numbers from 2 to 24, you will simply be taking half of 24. This will give you a number 12. Now, 12 should be multiplied with its immediate next number, i.e. 13. Thus, if you add all the even numbers from 2 to 24, you will get the sum as 156 (12 x 13). Is it not a shortcut method?

Let us try to find out short cut method for adding sequence of odd numbers beginning from 1 to a two digit odd number. Suppose we want to find out sum of odd numbers from 1 through 33. Our original idea of adding this sequence will give us in all 17 pairs (to recollect read article adding sequential numbers (II)). Each of these pairs will have to numbers which add to 34. This total we would know by adding the first and the last numbers in the sequence. Now the sum of the sequence is half of (34 × 17). This can be solved by multiplying half of 34 by 17. This is nothing but 17 × 17 = 289. So, the sum of all the odd numbers from 1 through 33 is 289.

Now, take some more sequences of odd numbers beginning from 1 through any two digit number and try to find out the pattern in the procedure. First thing you will find that you have always two numbers to multiply in the end. Secondly you will find that both the numbers you multiply are the same. Thus, you have to find out a single number which is to be squared to get the sum of consecutive odd numbers. The single number that you need to know can be obtained by adding 1 to the last two digit odd number in the sequence and then dividing it by 2.

Now try adding a sequence of odd numbers from 1 to 67. Adding 1 to 67, your sum will be 68. Half of 68 are 34. Square 34 are 1156. Thus sum of all the consecutive odd numbers from 1 to 67 is 1156.