Algebra Arithmetic Math Olympiad PRMO

Missing Integers | PRMO II 2019 | Question 1

Try this beautiful problem from the Pre-RMO II 2019, based on Missing Integers. You may use sequential hints to solve the problem.

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



Check the Answer

But try the problem first…

Answer: is 78.

Suggested Reading

PRMO II, 2019, Question 1

Elementary Algebra by Hall and Knight

Try with Hints

First hint


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.

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