1. ## subsequence

(an)=(2n-1) is a sequence.
(an+1) and (a2n) are subsequences of (an).
Is the following statement true?
"Sum of the subsequences is also a subsequence."

According to me this is wrong. According to the book true.
When we put n+1 in place of n we get 2n+1
When we put 2n in place of n we get 4n-1
If we sum 2n+1+4n-1=6n.So the sum is not a subsequence.

2. ## Re: subsequence

Originally Posted by kastamonu
(an)=(2n-1) is a sequence.
(an+1) and (a2n) are subsequences of (an).
Is the following statement true?
"Sum of the subsequences is also a subsequence."

According to me this is wrong. According to the book true.
When we put n+1 in place of n we get 2n+1
When we put 2n in place of n we get 4n-1
If we sum 2n+1+4n-1=6n.So the sum is not a subsequence.
$a_{2n}=\{-1,3,7,11\cdots\}.$ Starting with $n=0$
In order to be a subsequence each term must be in the sequence an the order must be preserved.
Are all of those is the original sequence? Are they in the same order?

3. ## Re: subsequence

an=1,3,5.......
a2n=4n-1=3,7......
They are in the same order. But as far as I know we can't start with 0.Numbers must be counting numbers not natural numbers.

4. ## Re: subsequence

Originally Posted by kastamonu
an=1,3,5.......
a2n=4n-1=3,7......
They are in the same order. But as far as I know we can't start with 0.Numbers must be counting numbers not natural numbers.
You do not count with zero? What primitive society do you live in?

5. ## Re: subsequence

$(a_n)$ is the positive odd numbers

I'm assuming that $(a_{n+1})_n = a_{n+1}$, this is also a sequence of odd numbers

Finally $(a_{2n})$ which I assume means $(a_{2n})_n = a_{2n}$ is also a sequence of odd numbers.

Their sum will be element by element a sum of two odd numbers which is even.

Thus their sum is a sequence of even numbers.

Their sum is not a sub-sequence of $(a_n)$

7. ## Re: subsequence

I remember a similar problem from when I took analysis. It said, given two sequences $a_n,b_n$ and any subsequences $(a_i)_{i\in I}, (b_j)_{j\in J}$ where $I,J \subseteq \mathbb{Z}^+$

Then the sum of the subsequences is a subsequence of the sum of the full sequences. That is as close as I can think to a similar statement that your book would say is true.

8. ## Re: subsequence

It is true but for this problem it is wrong.

9. ## Re: subsequence

Originally Posted by kastamonu
It is true but for this problem it is wrong.
That's because I stated the theorem wrong... It has been so many years since I took analysis... I'll try to find the actual statement.