Need help solving a question related to arithmetic progression with logarithm after it

Question

The question is as follows:

In the sequence,

$$ (a_{1}, a_{2}, a_{3}, ..., a_{9}, a_{10}) $$

There is a AP in which,

$$ a_2+a_9=243 $$

Resolve,

$$ \log_3(a_1+a_{10})^2 $$

I can do basic AP's and GP's without sweating, however, I really don't know how to answer this one.

Answer 1

Because it's an arithmetic progression, $a_1+a_{10}=a_2+a_9$,

so $\log_3(a_1+a_{10})^2=2\log_3(243)=2\times5=10$.

Answer 2

Hint: If the difference for the progression is $d$, then $$a_2+a_9 = \underbrace{a_2 -d}_{a_1}+\underbrace{d+a_9}_{a_{10}}$$