Finding $m$ given an arithmetic sequence

Question

Suppose we have an arithmetic sequence $\{a_n\}$ so that it has $m$ terms. Given that $m$ is odd, the sum of the odd index terms $(a_1, a_3, \cdots)$ is $44,$ and the sum of the even index terms $(a_2,a_4,\cdots)$ is $33,$ find the value of $m.$

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First, I noted that the last addend of the odd index term sum must be $a_m,$ since $m$ is odd. However, from here, I didn't really have any idea where to go because setting $a$ equal to the first term and $d$ being the common difference seemed too complicated. Can someone help please?

Answer 1

Note there are $\frac{m+1}{2}$ odd index terms and $\frac{m-1}{2}$ even index terms.

These sequences separately form AP with common difference $2d$. So

$$ \dfrac{\tfrac{m+1}{2}(a_1+a_m) }{\tfrac{m-1}{2}(a_2+a_{m-1}) }=\dfrac{44}{33} $$

And $a_1+a_m=a_2+a_{m-1}$