1996 AHSME Problems/Problem 24
Problem
The sequence consists of ’s separated by blocks of ’s with ’s in the block. The sum of the first terms of this sequence is
Solution
The sum of the first numbers is
The sum of the next numbers is
The sum of the next numbers is
In genereal, we can write "the sum of the next numbers is ", where the word "next" follows the pattern established above.
Thus, we first want to find what triangular numbers is between. By plugging in various values of into , we find:
Thus, we want to add up all those sums from "next number" to the "next numbers", which will give us all the numbers up to and including the number. Then, we can manually tack on the remaining s to hit .
We want to find:
Thus, the sum of the first terms is . We have to add more s to get to the term, which gives us , or option .
Note: If you notice that the above sums form , the fact that appears at the end should come as no surprise.
See also
1996 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 23 |
Followed by Problem 25 | |
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