Missing Integers | PRMO II 2019 | Question 1

Consider the sequence of numbers [n+\(\sqrt{2n}+\frac{1}{2}\)] for \(n \geq 1\), where [x] denotes the greatest integer not exceeding x. If the missing integers in the sequence are \(n_1<n_2<n_3<...\) then find \(n_{12}\).

\([n+\sqrt{2n}+\frac{1}{2}]\)=[\((\sqrt{n}+\frac{1}{\sqrt{2}})^2\)]

Let P=[\((\sqrt{n}+0.7)^2\)]

given \(n \geq 1\), put n=1 gives P=2

n=2 gives P=4

n=3 gives P=5

n=4 gives P=7

n-5 gives P=8

n=6 gives P=9

n=7 gives P=11

here missing number are

1,3,6,10,... which is following a certain pattern

1, 1+2, 3+3, 6+4, 10+5, 15+6, 21+7, 28+8, 36+9, 45+10, 55+11, 66+12.

so, \(n_{12}\)=78.

Subscribe to Cheenta at Youtube