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 ?
Number Theory, Sequences
[/et_pb_accordion_item][et_pb_accordion_item title="Difficulty Level" _builder_version="4.0" open="off"]7/10
[/et_pb_accordion_item][et_pb_accordion_item title="Suggested Book" _builder_version="3.29.2" open="off"]Challenges and Thrills of Pre-College Mathematics[/et_pb_accordion_item][/et_pb_accordion][et_pb_text _builder_version="3.27.4" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#e2e2e2" background_color="#0c71c3" custom_margin="48px||117px|||" custom_padding="20px|20px|20px|20px" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3"]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 ?
[/et_pb_tab][et_pb_tab title="Hint 2" _builder_version="4.0"]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 ?
[/et_pb_tab][et_pb_tab title="Hint 3" _builder_version="4.0"]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 ?
[/et_pb_tab][et_pb_tab title="Hint 4" _builder_version="4.0"]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 !
[/et_pb_tab][/et_pb_tabs][et_pb_text _builder_version="3.27.4" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#e2e2e2" background_color="#0c71c3" custom_margin="-14px||48px|||" custom_padding="20px|20px|20px|20px" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3"]Math Olympiad is the greatest and most challenging academic contest for school students. Brilliant school students from over 100 countries participate in it every year. Cheenta works with small groups of gifted students through an intense training program. It is a deeply personalized journey toward intellectual prowess and technical sophistication.[/et_pb_blurb][et_pb_button button_url="https://cheenta.com/matholympiad/" url_new_window="on" button_text="Learn More" button_alignment="center" _builder_version="3.23.3" custom_button="on" button_bg_color="#0c71c3" button_border_color="#0c71c3" button_border_radius="0px" button_font="Raleway||||||||" button_icon="%%3%%" background_layout="dark" button_text_shadow_style="preset1" box_shadow_style="preset1" box_shadow_color="#0c71c3"][/et_pb_button][et_pb_text _builder_version="3.27.4" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#e2e2e2" background_color="#0c71c3" custom_margin="50px||50px" custom_padding="20px|20px|20px|20px" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3"]
