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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

Real Numbers

Algebra

Integers

## Check the Answer

But try the problem first…

Source

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.