let ![\[x_n=p_1+\cdots+p_n\]](https://latex.artofproblemsolving.com/2/3/c/23c630cbd7a08b8f272a499c1ba6dd1c2368aead.png)
where 
are the first primes. Prove that for each positive integer , there is an integer 
such that 
[/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[/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" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3" custom_margin="48px||48px" custom_padding="20px|20px|20px|20px" _i="1" _address="0.1.0.1"] , 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"]
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%%" 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"]
