Understand the problem

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   $x_n<k_n^2<x_{n+1}$
Source of the problem
SMO (senior)-2014 stage 2 problem 4

Topic
Number Theory
Difficulty Level
Medium
Suggested Book
Excursion in Mathematics

Start with hints

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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 \( \forall n \geq 5 \) .
Observe  \( \forall n \geq 5 \) we have \( P_n > (2n-1) \). [where \( n \in Z^+ \)] Then try to use  \( x_n =  P_1 + P_2 + …+P_5+….. +P_n  > 1 +3 + ….+ 9 +… (2n-1) = n^2 \\  \Rightarrow x_n > n^2  , \forall n \geq 5[where \ n \in Z^+]   \) .
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 .

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 ,\( (n+m+1)^2 \in (x_n , x_{n+1}) \)  .    

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