My little sister is working on geometric and arithmetic sequences and writing rules for them. She’s using formulas I have NEVER seen before nor can find anywhere on the Internet. The formula uses a t(n) term and an a_{0} term. They look something like this:

ARITHMETIC: t(n)=d^{n}+a_{0}

GEOMETRIC: t(n)=a_{0}r^{n}

Can someone explain how this new formula works and why my little sister is being taught this new one instead of the former (which makes more sense to me)? I don’t understand how the t(n) one works.

Perhaps you should tell us what "the former" is, so we can be sure what we're talking about.

In each case, a

_{0} is the "first" (actually, zeroth) term, which you start at (for n=0). Then, t(n) is the nth term, which you are calculating for any non-negative integer n. It could also have been called a

_{n}.

In an arithmetic sequence, you get there by adding the common difference, d, n times, which amounts to adding n times d.

In a geometric sequence, you multiply by the common ratio, r, n times, which amounts to multiplying by r

^{n}.

In each case, t(0) = a

_{0}, as expected.

Not knowing what you are comparing these to, my guess is that you are used to starting with index 1 rather than 0. That makes the formulas more complicated, which is the reason for starting with 0 instead.