# I don’t understand the new formula my little sis is using for arithmetic/geometric sequences

#### smoochiepook

##### New member
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 a0 term. They look something like this:

ARITHMETIC: t(n)=dn+a0
GEOMETRIC: t(n)=a0rn

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.

#### MarkFL

##### Super Moderator
Staff member
Hello, and welcome to FMH!

I would write:

AP: $$\displaystyle a_n=a_0+dn$$

GP: $$\displaystyle a_n=a_0r^n$$

These are formulas for the $$n$$th term in the respective sequences.

#### Dr.Peterson

##### Elite Member
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 a0 term. They look something like this:

ARITHMETIC: t(n)=dn+a0
GEOMETRIC: t(n)=a0rn

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, a0 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 an.

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 rn.

In each case, t(0) = a0, 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.

#### pka

##### Elite Member
ARITHMETIC: t(n)=dn+a0
GEOMETRIC: t(n)=a0rn
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.
I would write: AP: $$\displaystyle a_n=a_0+dn$$
GP: $$\displaystyle a_n=a_0r^n$$
Thank you Mark for the corrections to the OP.
I would point out that the first term of each occurs when $$\displaystyle n=0$$ that is the first term is $$\displaystyle a_0$$.
This is tricky because the first term is when $$\displaystyle n=0$$ not when $$\displaystyle n=1$$.
For that reason many textbooks will say that $$\displaystyle a_0+d\cdot(n-1)$$ is the nth term of an AP.

#### Harry_the_cat

##### Senior Member
Thank you Mark for the corrections to the OP.
I would point out that the first term of each occurs when $$\displaystyle n=0$$ that is the first term is $$\displaystyle a_0$$.
This is tricky because the first term is when $$\displaystyle n=0$$ not when $$\displaystyle n=1$$.
For that reason many textbooks will say that $$\displaystyle a_0+d\cdot(n-1)$$ is the nth term of an AP.
Should the subscript on a be 1 not 0 in your last expression?