Part (a) of the question asks to calculate the values of a_2, a_3, a_4, a_5, a_6, a_7, and a_8 for the sequence. Part (b) modifies the sequence to start with a_1 = k, where k is any positive integer, and the same recursion formula applies: for n ≥ 2, a_n = S(a_(n-1)). The question notes that for many choices of k, the sequence a_n will eventually reach and remain at 1, but this is not always the case. It asks to find, with an explanation, two specific values of k for which the sequence a_n never reaches 1.
The question describes a function S(k) which is defined as the sum of the positive divisors of a positive integer k, minus k itself. The function S(1) is defined as 1, and for any positive integer k greater than 1, S(k) is calculated as S(k) = σ(k) - k, where σ(k) is the sum of all positive divisors of k.
Some examples of S(k) are given:
- S(1) = 1
- S(2) = 1
- S(3) = 1
- S(4) = 3
- S(5) = 1
- S(6) = 6
- S(7) = 1
- S(8) = 7
- S(9) = 4
The question then introduces a recursive sequence a_n with the following rules:
- a_1 = 12
- For n ≥ 2, a_n = S(a_(n-1))
Part (a) of the question asks to calculate the values of a_2, a_3, a_4, a_5, a_6, a_7, and a_8 for the sequence.
Part (b) modifies the sequence to start with a_1 = k, where k is any positive integer, and the same recursion formula applies: for n ≥ 2, a_n = S(a_(n-1)). The question notes that for many choices of k, the sequence a_n will eventually reach and remain at 1, but this is not always the case. It asks to find, with an explanation, two specific values of k for which the sequence a_n never reaches 1.
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