Try this beautiful problem from the PRMO II, 2019 based on Missing Integers.

## Missing Integers – PRMO II 2019

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 \(n_{12}\).

- is 107
- is 78
- is 840
- cannot be determined from the given information

**Key Concepts**

Real Numbers

Algebra

Integers

## Check the Answer

But try the problem first…

Answer: is 78.

PRMO II, 2019, Question 1

Elementary Algebra by Hall and Knight

## Try with Hints

First hint

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

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

Second Hint

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

Final Step

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.

## Other useful links

- https://www.cheenta.com/smallest-perimeter-of-triangle-aime-2015-question-11/
- https://www.youtube.com/watch?v=ST58GTF95t4&t=140s

Google