the first term of a sequence is one. the second term is two. the third term is $3$. the 4th term is $5$. the $5$th term is $7$.
all terms after the second term repeat and follow the order $3,5,7$; for example, the first $11$ numbers are $1,2,3,5,7,3,5,7,3,5,7$…
What patterns do the $5$th, and $7$th terms have? E.g. they're all multiples of n.
I want to find patterns for the $5$th and $7$th terms to solve for the $100$th and $101$th term.
Best Answer
Let $a_n$ be the terms of your sequence. Then for $n\ge 1$ we have: $$a_n=\begin{cases} 1 &\text{ if } n=1 \\2\,\,\,\,\,\,\,\,\,\,\,\,\ &\text{ if } n=2 \\3 &\text{ if }n\equiv0\pmod3 \\5 &\text{ if }n\equiv1\pmod3\text{ and } n\ge3 \\7 &\text{ if }n\equiv2\pmod3\text{ and } n\ge3 \end{cases}$$
Since $100\equiv 1\pmod3$ we have $a_{100}=5$ and therefore $a_{101}=7$.