I am right now in the Hebrew University Academic Prep School which in its level is equivalent to an American high school/college. It's a one-year academic prep school for engineering and exact science.
Due to the war in Israel right now, the academic year started a little bit later than usual. So far we have not learned much about arithmetic series, just the basics, including finding the general term of $a_n$, and finding the sum of $a_n$, (at even and odd places). I am including the formulas that we were taught so it can help you understand how to answer the question.
$S_n = \frac{(2a_1 + d(n-1))n}{2} = \frac{(a_1 + a_n)n}{2}$
$S_{2n_{even}} = \frac{(2a_2 + 2d(n-1))n}{2}$, $S_{2n_{odd}} = \frac{(2a_1 + 2d(n-1))n}{2}$
$S_{2n+1_{even}} = \frac{(2a_2 + 2d(n-1))n}{2}$, $S_{2n+1_{odd}} = \frac{(2a_1 + 2d(n))n}{2}$
$a_n = a_1 + d (n-1)$
$\begin{cases}
a_1 \\
a_{n+1} &= a_n + d
\end{cases}
$
Question:
given that $a_n$ is an arithmetic sequence, and $$a_{14}^2+a_{15}^2+a_{16}^2=35$$ $$a_{15} > 0$$ find the general term of an $a_n$
It would be greatly appreciated if you could solve this with the basic equations as we don't really know much about series. Thank you so much.
Choose $a_{14} = A-B$, $a_{15}=A$ and $a_{16}=A+B$, where $A>0$ so that the given condition can be written as $$3A^2+2B^2=35$$ Now there’s multiple (infinite) pairs of $A$ and $B$ that can be chosen to satisfy this equation. Each of those would correspond to different arithmetic sequences.
However, if we look over the integers we can reduce this quite easily: it’s clear that $|A|<4$ and that $|B|<5$, so that $(A,B)$ is $(3,2)$ or $(3,-2)$ or $(1,4)$ or $(1,-4)$. In the first case we get that the arithmetic sequence can be described as $a_n = 2n-27$, for the second that $a_n = 33-2n$, in the third that $a_n = 4n -59$ and in the fourth that $a_n = 61-4n$ where $n \in \mathbb Z^+$ and $a_n$ denotes the $n$th term of the arithmetic sequence.