# Understand the problem

A sequence of natural numbers is constructed by listing the first , then skipping one, listing the next , skipping , listing , skipping , and, on the th iteration, listing and skipping . The sequence begins . What is the th number in the sequence ?

##### Source of the problem

##### Topic

Number Theory, Sequences

##### Difficulty Level

7/10

##### Suggested Book

# Start with hints

Stuck…? Well, don’t worry. *The key to solving this problem is not thinking too much about the skips.* We can start with natural numbers, from **1** to **500,000**. So, a useful strategy could be to find how many numbers we have actually skipped, **n** and then add them back accordingly. So, now could you take things on from here ?

If you’re a tad bit doubtful of where we’re heading even now, well no problem. Clearly, we can say ** 999.(1000) / 2 < 500,000 < 1000.(1001) / 2** So, now can you find out how many blocks of gaps we have in the sequence ?

Now see, finding the blocks of gaps easy ! There’s just one small thing you would have to recall. **We began the count from 4…so now, the number of skipped blocks in the sequence = 999 – 3 = 996.** Now to find n, from the number of blocks , we have = **(996.997) / 2 = 496,506** This stands for the number of numbers we skipped. Now concluding this is fairly easy…could you try that out yourself ? * *

What remains for us to do is to add back those skipped numbers to the count, **500,000** to obtain the final answer. That gives us = **500000 +496506 = 996506**

And we are done !

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# Connected Program at Cheenta

#### Math Olympiad Program

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