Consider a set, S=[1,2,3, ldots . n] n ∈ N. A non empty subset of S is called ' k ' if the arithmetic means of its elements is a natural number. Let αn be the number of ' k ' subsets and β n be the number of all singleton subsets αn will be

Question

Consider a set, S=[1,2,3, ldots . n] n ∈ N. A non empty subset of S is called ' k ' if the arithmetic means of its elements is a natural number. Let αn be the number of ' k ' subsets and β n be the number of all singleton subsets αn will be (A) even if βn is even (B) odd if βn is even (C) even, irrespective if βn is even or odd (D) None of these

Tardigrade Admin · Accepted Answer

Correct answer is (a) even if βn is even

Q. Consider a set, $S = [1, 2, 3, \dots . n] n \in N$ . A non empty subset of $S$ is called ' $k$ ' if the arithmetic means of its elements is a natural number. Let $α_{n}$ be the number of ' $k$ ' subsets and $β_{n}$ be the number of all singleton subsets
$α_{n}$ will be

even if $β_{n}$ is even

odd if $β_{n}$ is even

even, irrespective if $β_{n}$ is even or odd

None of these

Correct answer is (a) even if $β_{n}$ is even

Q. Consider a set, S=[1,2,3,….n]n∈N. A non empty subset of S is called ' k ' if the arithmetic means of its elements is a natural number. Let αn​ be the number of ' k ' subsets and βn​ be the number of all singleton subsets αn​ will be

even if βn​ is even

odd if βn​ is even

even, irrespective if βn​ is even or odd