Question

what is the sum of all 2 digit numbers from 10 to 99

Answer 1

\[\sum _{n=10}^{99} n=4905 \]

Answer 2

Imagine you've written out all the numbers from 10 to 99 in a long line:

10 11 12 13 14 15 16 17 18 19 ... 98 99

now underneath write them again, except in the opposite order:
99 98 97 96 95 94 93 92 91 90 ... 11 10

If you add down the columns, you'll see that each column adds to 10 + 99 = 109

So we can figure out the sum by multiplying 109 * the number of numbers from 10 to 99 and dividing the result by 2.

To find the number of numbers from 10 to 99, observe that the number of numbers from 10 to 11 is 11-10 +1 = 2 (10 and 11). The number of numbers from 10 to 12 is 12-10 + 1 = 3 (10, 11, 12). So, our formula for the sum of a numbers from any starting point to any ending point is:

(ending number - starting number + 1) * (starting number + ending number) * (1/2).

If we plug in the numbers for this problem:

\[(99-10+1)*(10+99)*\frac{1}{2} = \frac{1}{2}*90*109 = 4905\]

Answer 3

@whpalmer4

Yes,\[\frac{99*100}{2}-\frac{9*10}{2}=4905 \]Using Mathemaitca it was a matter of clicking on the Basic Math Palette's sigma boiler plate and then, n tab 10 tab 99 tab n numeric_key_pad_enter , Computation time was 63 microseconds. Less mental exertion.

Thank you anyway for your analysis.

Answer 4

Yeah, but the chap (or chapette) asking the question needs to know how to do it without Mathematica, right?

\(56.99999999819738 ~\mu s\) on my system :-)

Answer 5

Interesting timing. What hardware were you running on? Mine is a mid 2010 Applie IMac with 8 gb of ram.

Answer 6

Sorry for the spelling errors tonight.

Answer 7

I've got the iMac 11,3 from late 2010, 2.97 GHz i7. I'm not attaching too much significance to the digits beyond the first two in that timing number!

Answer 8

Thank you for the conversation. Have to bail out for dinner now.

Answer 9

Likewise! 'Til next time...