Difference between revisions of "1985 AJHSME Problem 2"
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We know that <math>10(90) = 900</math> and <math>10(100) = 1000.</math> Quick estimation reveals that this sum is in between these two numbers, so the only answer available is <math>\text{(B)}.</math> | We know that <math>10(90) = 900</math> and <math>10(100) = 1000.</math> Quick estimation reveals that this sum is in between these two numbers, so the only answer available is <math>\text{(B)}.</math> | ||
==Solution 3== | ==Solution 3== | ||
| − | You see that <math>90+91+92+93+94+95+96+97+98+99</math> is equal to <math>0+1+3+4+5+6+7+8+9+900. You can use the formula | + | You see that <math>90+91+92+93+94+95+96+97+98+99</math> is equal to <math>0+1+3+4+5+6+7+8+9+900</math>. You can use the formula $\frac{n(n+1)}{2} to get 45+900=945, which is B. |
==Video Solution== | ==Video Solution== | ||
Revision as of 01:20, 22 March 2025
Problem
Solution 1
We can add as follows:
![\[90+91+92+93+94+95+96+97+98+99= 10(90) +1+2+3+4+5+6+7+8+9 = 900 + 45 = \boxed{945}\]](http://latex.artofproblemsolving.com/1/d/b/1db4e2342c772b4aed18d27770793629634fa8d5.png)
The answer is 
Solution 2
Pair the numbers like so:
![\[(90+99)+(91+98)+(92+97)+(93+96)+(94+95)\]](http://latex.artofproblemsolving.com/9/0/e/90e6ffd152b0189a5f7fd1521ff17296c91172b9.png)
The sum of each pair is 
and there are 
pairs, so the sum is 
and the answer is
Cheap Solution
We know that 
and 
Quick estimation reveals that this sum is in between these two numbers, so the only answer available is
Solution 3
You see that 
is equal to 
. You can use the formula $\frac{n(n+1)}{2} to get 45+900=945, which is B.
Video Solution
~savannahsolver
