I'm about to start college and I've been sent some documents about what we will be doing in the first classes. One of these is the theory about summation and product of sequences, and some problems. I've read the theory and wanted to do the first problem, but I ran into a lot of trouble. (I'm not studying in English so I have to translate, but I'll do my best).
The problem says:
For certain numbers $a_i$, i $\ge1$: $$\sum_{i=1}^n a_i = n^3.$$ What is the value of: $$\sum_{i=5}^{12} a_i$$ And what is the value of: $a_{21}$
To start with this problem seemed very werid to me, since $a_{21}$ doesn't exist (21<12). I supposed that whoever wrote the problem wanted to write $a_{12}$(There are other mistakes like this on the document). Then "For certain numbers $a_i$" makes me think that $a_i$ is a constant number so that: $$\sum_{i=m}^{n} k=n^3$$ (The problem says that $i\ge1$, and them Im asked when $i=5$, so I replaced $5$ for $m$, and $a_i$ for $k$).Then what the summation is doing is sum $k$ (n-m+1) times.
This gives me the equation: $$k(n-m+1)=n^3$$ And replacing the numbers with $n=12, m=5$, I get that $k=288=2·12^2$.
Since the summation does this: $k_5$+$k_6$+$k_7$+$k_8$+$k_9$+$k_{10}$+$k_{11}$+$k_{12}$, it can be expressed as $6k$, so: $$6k=n^3$$ $$6·2·12^2=12^3$$ $$12^3=12^3$$ Which is true and matches $\sum_{i=m}^{n} k=n^3$.
I'd be happy with that, but then I'm asked for $a_{12}$ which I supposed it's a constant and I ask myself If I'm doing the whole thing wrong and I just didn't understand the question.
Sorry for any inconsistencies with English or the way I wrote the expressions.
Given that $$\sum_{i=1}^n a_i = n^3$$
we have that $$\sum_{i=1}^{21} a_i = 21^3$$
and $$\sum_{i=1}^{20} a_i = 20^3$$
Hence, $$a_{21} = \sum_{i=1}^{21} a_i - \sum_{i=1}^{20} a_i = 21^3 - 20^3$$