In Singapore, there is this competition named SMO. Recently, they released some example questions. The following is one of them...
Let $a_1, a_2, a_3$ be an arithmetic progression with $a_1$>0 and $3a_8=5a_{13}$. Let $S_n=a_1+a_2+...+a_n$ for all positive integer $n$. Find the integer $n$ such that $S_n$ has the largest possible value.
For this question, I need help in understanding the formula. Anyone help?
Hint
For an arithmetic progression, you have $$a_n=a_1+(n-1) r$$ So $3a_8=5a_{13}$ write $$3a_1+3(8-1) r=5a_1+5(13-1)r$$ This, once developed, gives you the relation between $r$ and $a_1$.
Can you take from here ?