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tmua.me Extreme Set B Paper 2 · Q13

Paper 2Logic & proof

For a real constant kk with 0≤k≤20\leq k\leq2, define

an=n+(−1)nkn(n=1,2,3,…).a_n=n+\frac{(-1)^n k}{n} \qquad(n=1,2,3,\ldots).

Consider the statement:

For every real number ε>0\varepsilon>0, there exist distinct positive integers mm and nn such that

∣am−an∣<ε.|a_m-a_n|<\varepsilon.

For which values of kk is this statement true?

Ak=65k=\dfrac65 only
Bk=1k=1 or k=65k=\dfrac65
C1≤k≤21\leq k\leq2
D65≤k≤2\dfrac65\leq k\leq2
EEvery kk with 0≤k≤20\leq k\leq2
FThere are no such values of kk

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