Let an = least power of 2 that divides n. Then isa)divergent to infinityb)boundedc)having a subsequence converging to 3d)convergentCorrect answer is option 'B'. Can you explain this answer?

Question

Let an = least power of 2 that divides n. Then<an> isa)divergent to infinityb)boundedc)having a subsequence converging to 3d)convergentCorrect answer is option 'B'. Can you explain this answer?

Answer

Here actually the range set is {0,0,1,0,1,0,1,..}

so the set has two limit points hence it is not convergent

but clearly it is bounded

Answer

The sequence {an} is defined as follows:

a1 = 1
a2 = 2
a3 = 1
a4 = 4
a5 = 1
a6 = 2
a7 = 1
a8 = 8
a9 = 1
a10 = 2
...

In this sequence, an is the least power of 2 that divides n. For example, a5 = 1 because 5 is not divisible by 2, a6 = 2 because 6 is divisible by 2 but not by 4, and a8 = 8 because 8 is divisible by 2 and by 4, but not by 16.

The sequence can be generated by iterating through all positive integers and finding the smallest power of 2 that divides each number. This can be done by dividing the number by 2 repeatedly until the result is not divisible by 2, and then taking the last power of 2 used.

The sequence {an} has several interesting properties. For example, the sum of the first n terms of the sequence is equal to n. This can be proven by induction. Additionally, every positive integer can be expressed as a product of powers of 2, and the sequence {an} represents the prime factorization of each number.

Overall, the sequence {an} provides a useful way to analyze the divisors of positive integers and their prime factorization.

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