Question

What is d of the arithmetic sequence for which a5=24 and a9=40?

Answer 1

In arithmetic sequences, we know that they always differ by the same amount. That mean we can clearly construct them by incrementing our common differences. Let $a_1=s$, that means going from 1 to 5, 1 non inclusive, would give us 4 more common differences, and going from 5 to 9, 5 non inclusive, would give us 4 again. That means:

$a_5=s+4d\\ a_9=s+8d$

Giving a_5 and a_9 numerical values then solving for d:

$24=s+4d\\ 40=s+8d \implies\\ 16=4d \implies\\ d=4$

I subtracted the top equation from the bottom one if that isn't clear. With that, we are done.