Question

Write a formula for the an in each of the following sequences {an}∞n=1 given by (i) 2,1,4,3,6,5,8,7,... (ii) 1, 3, 6,10,15,... (iii) 1, −4,9,−16,25,−36,... Which ones among these are the subsequence of {n}∞n=1?

Answer 1

1) a_n = n - 1, for n=2$i$ (even), $i\in\N$.

a_n = n + 1, for n=2$i$-1 (odd), $i\in\N$.

2) a_n = $\frac{n(n+1)}{2}$ .

3) a_n = - n², for n=2$i$ (even), $i\in\N$.

a_n = n², for n=2$i$-1 (odd), $i\in\N$.

{n} = 1, 2, 3, 4, 5...n

1) This sequence can be obtained from {n} without removing members, but with changing the order of the members. She's not a subsequence {n}.

2) This sequence can be obtained from {n} with removing members, without changing the order of the members. She's a subsequence {n}.

3) This sequence cannot be obtained from {n}. She is divergent(dispersent).

Answer: 2.