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SMO (senior) -2014/problem-4 Number Theory

[et_pb_section fb_built="1" _builder_version="3.22.4" fb_built="1" _i="0" _address="0"][et_pb_row _builder_version="3.25" _i="0" _address="0.0"][et_pb_column type="4_4" _builder_version="3.25" custom_padding="|||" custom_padding__hover="|||" _i="0" _address="0.0.0"][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" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3" custom_padding="20px|20px|20px|20px" _i="0" _address="0.0.0.0"]

Understand the problem

[/et_pb_text][et_pb_text _builder_version="3.27.4" text_font="Raleway||||||||" background_color="#f4f4f4" box_shadow_style="preset2" custom_margin="10px||10px" custom_padding="10px|20px|10px|20px" _i="1" _address="0.0.0.1"]For each positive integer $n$ let  \[x_n=p_1+\cdots+p_n\]  where  $p_1,\ldots,p_n$   are the first $n$ primes. Prove that for each positive integer $n$, there is an integer $k_n$ such that   $x_n<k_n^2<x_{n+1}$[/et_pb_text][/et_pb_column][/et_pb_row][et_pb_row _builder_version="3.27" _i="1" _address="0.1"][et_pb_column type="4_4" _builder_version="3.25" custom_padding="|||" custom_padding__hover="|||" _i="0" _address="0.1.0"][et_pb_accordion open_toggle_text_color="#0c71c3" _builder_version="3.27" toggle_font="||||||||" body_font="Raleway||||||||" text_orientation="center" custom_margin="10px||10px" _i="0" _address="0.1.0.0"][et_pb_accordion_item title="Source of the problem" open="on" _builder_version="3.27" hover_enabled="0" _i="0" _address="0.1.0.0.0"]SMO (senior)-2014 stage 2 problem 4

[/et_pb_accordion_item][et_pb_accordion_item title="Topic" _builder_version="3.27" hover_enabled="0" _i="1" _address="0.1.0.0.1" open="off"]Number Theory[/et_pb_accordion_item][et_pb_accordion_item title="Difficulty Level" _builder_version="3.27" hover_enabled="0" _i="2" _address="0.1.0.0.2" open="off"]Medium[/et_pb_accordion_item][et_pb_accordion_item title="Suggested Book" _builder_version="3.27" hover_enabled="0" _i="3" _address="0.1.0.0.3" open="off"]Excursion in Mathematics

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Start with hints

[/et_pb_text][et_pb_tabs active_tab_background_color="#0c71c3" inactive_tab_background_color="#000000" _builder_version="3.27.4" tab_text_color="#ffffff" tab_font="||||||||" background_color="#ffffff" hover_enabled="0" _i="2" _address="0.1.0.2"][et_pb_tab title="Hint 0" _builder_version="3.27" hover_enabled="0" _i="0" _address="0.1.0.2.0"]Do you really need a hint? Try it first!

[/et_pb_tab][et_pb_tab title="Hint 1" _builder_version="3.27.4" hover_enabled="0" _i="1" _address="0.1.0.2.1"]We have \( P_1 = 2 , P_=3 , P_3=5 , P_4 =7 , P_5 = 11 \ and \ so \ on .... \). Now to understand the expression   $x_n<k_n^2<x_{n+1}$  ,  observe .   \( For \ n=1 \ ,  \ 2 < 2^2 < 2+3 \)  \( For \ n=2 \ ,  \ 2+3 < 3^2 < 2+3+5 \) \( For \ n=3 \ ,  \ 2+3+5 < 4^2 < 2+3 +5+7\) \( For \ n=4 \ ,  \ 2+3 +5+7 < 5^2 <2+3 +5+7 +11 \) Now proceed to prove \( \forall n \geq 5 \) .[/et_pb_tab][et_pb_tab title="Hint 2" _builder_version="3.27.4" hover_enabled="0" _i="2" _address="0.1.0.2.2"]Observe  \( \forall n \geq 5 \) we have \( P_n > (2n-1) \). [where \( n \in Z^+ \)] Then try to use  \( x_n =  P_1 + P_2 + ...+P_5+..... +P_n  > 1 +3 + ....+ 9 +... (2n-1) = n^2 \\  \Rightarrow x_n > n^2  , \forall n \geq 5[where \ n \in Z^+]   \) .[/et_pb_tab][et_pb_tab title="Hint 3" _builder_version="3.27.4" hover_enabled="0" _i="3" _address="0.1.0.2.3"]Think if \( x_n=P_1 + P_2 + .... + P_5 +...P_n = b^2 for \ some \ n ,  b \in Z^+ \) , then we are done . If not so , then think \( m \) be the largest non negative integer such that  \( (n+m)^2 < x_n \) . Now note that the next perfect square is \( (n+m+1)^2  \) . Observe that if we can prove that   \( (n+m+1)^2 - (n+m)^2 = (2n+ 2m +1) \geq P_{n+1} \)  , then we are done . Now try to verify this claim .[/et_pb_tab][et_pb_tab title="Hint 4" _builder_version="3.27.4" hover_enabled="0" _i="4" _address="0.1.0.2.4"]

Suppose our claim is not true  i.e. \( P_n < 2n + 2m +1\) . So,   \( P_n < 2n + 2m +1 \\ \Rightarrow 2n+ 2m \geq P_n , \forall n \in Z^+ \\ \Rightarrow (2n +2m-2)+(2n+ 2m -4)+.....2m \geq P_n  + P_{n-1}+......+P_1 \\ \Rightarrow n^2 + 2mn -n \geq P_n  + P_{n-1}+......+P_1  \\ \Rightarrow n^2 + 2mn -n \geq x_n \\ \Rightarrow  n^2 + 2mn +m^2 > n^2 + 2mn -n\geq x_n \\ \Rightarrow (n+m)^2 > x_n  \) .  Contradiction!  since we have assumed \( x_n = P_1  + P_2+......+P_{n-1} > (n+m)^2 \) . Thus ,\( (n+m+1)^2 \in (x_n , x_{n+1}) \)  .     [/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" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3" min_height="12px" custom_margin="50px||50px" custom_padding="20px|20px|20px|20px" _i="3" _address="0.1.0.3"]

Connected Program at Cheenta

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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://www.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%%" button_text_shadow_style="preset1" box_shadow_style="preset1" box_shadow_color="#0c71c3" background_layout="dark" _i="5" _address="0.1.0.5"][/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" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3" custom_margin="50px||50px" custom_padding="20px|20px|20px|20px" _i="6" _address="0.1.0.6"]

Similar Problems

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