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.

Source
Suggested Reading

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.

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