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

[smoochiepook]

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]

Hello, and welcome to FMH!

I would write:

AP: [MATH]a_n=a_0+dn[/MATH]
GP: [MATH]a_n=a_0r^n[/MATH]
These are formulas for the \(n\)th term in the respective sequences.

 

[Dr.Peterson]

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 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 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]

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: [MATH]a_n=a_0+dn[/MATH]GP: [MATH]a_n=a_0r^n[/MATH]

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]

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?