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

Do you really need a hint? Try it first!

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

Connected Program at Cheenta

Math Olympiad Program

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.

Similar Problems

Coin Toss Problem | AMC 10A, 2017| Problem No 18

Try this beautiful Problem on Probability from AMC 10A, 2017. Problem-18, You may use sequential hints to solve the problem.

GCF & Rectangle | AMC 10A, 2016| Problem No 19

Try this beautiful Problem on Geometry on Rectangle from AMC 10A, 2010. Problem-19. You may use sequential hints to solve the problem.

Fly trapped inside cubical box | AMC 10A, 2010| Problem No 20

Try this beautiful Problem on Geometry on cube from AMC 10A, 2010. Problem-20. You may use sequential hints to solve the problem.

Measure of angle | AMC 10A, 2019| Problem No 13

Try this beautiful Problem on Geometry from AMC 10A, 2019.Problem-13. You may use sequential hints to solve the problem.

Sum of Sides of Triangle | PRMO-2018 | Problem No-17

Try this beautiful Problem on Geometry from PRMO -2018.You may use sequential hints to solve the problem.

Recursion Problem | AMC 10A, 2019| Problem No 15

Try this beautiful Problem on Algebra from AMC 10A, 2019. Problem-15, You may use sequential hints to solve the problem.

Roots of Polynomial | AMC 10A, 2019| Problem No 24

Try this beautiful Problem on Algebra from AMC 10A, 2019. Problem-24, You may use sequential hints to solve the problem.

Set of Fractions | AMC 10A, 2015| Problem No 15

Try this beautiful Problem on Algebra from AMC 10A, 2015. Problem-15. You may use sequential hints to solve the problem.

Indian Olympiad Qualifier in Mathematics – IOQM

Due to COVID 19 Pandemic, the Maths Olympiad stages in India has changed. Here is the announcement published by HBCSE: Important Announcement [Updated:14-Sept-2020]The national Olympiad programme in mathematics culminating in the International Mathematical Olympiad...

Positive Integers and Quadrilateral | AMC 10A 2015 | Sum 24

Try this beautiful Problem on Rectangle and triangle from AMC 10A, 2015. Problem-24. You may use sequential hints to solve the problem.