# Understand the problem

##### Source of the problem

##### Topic

##### Difficulty Level

##### Suggested Book

# Start with hints

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.