1. ## Basic proof.

Proof that the rth term can be found by finding the difference between the sum of the first r terms and the sum of the first (r-1) terms: That is, $\displaystyle U_{r} = S_{r}-S_{r-1}$.

What I really don't understand is $\displaystyle S_{r-1}$. For example, what is the $\displaystyle S_{r-1}$ of $\displaystyle 3+5+7+9$? Is it $\displaystyle 0+3+5+7$? Or $\displaystyle 1+3+5+7$?

2. $\displaystyle S_{r}$ is the sum of terms up to and including $\displaystyle U_{r}$
$\displaystyle S_{r-1}$ is the sum of terms up to and including $\displaystyle U_{r-1}$

i.e.
$\displaystyle S_{r}=U_{1}+U_{2}+U_{3}+...+U_{r-1}+U_{r}$ -----(1)
$\displaystyle S_{r-1}=U_{1}+U_{2}+U_{3}+...+U_{r-2}+U_{r-1}$ -----(2)

(1)-(2): $\displaystyle U_{r}=S_{r}-S_{r-1}$

3. Originally Posted by acc100jt
$\displaystyle S_{r}$ is the sum of terms up to and including $\displaystyle U_{r}$
$\displaystyle S_{r-1}$ is the sum of terms up to and including $\displaystyle U_{r-1}$

i.e.
$\displaystyle S_{r}=U_{1}+U_{2}+U_{3}+...+U_{r-1}+U_{r}$ -----(1)
$\displaystyle S_{r-1}=U_{1}+U_{2}+U_{3}+...+U_{r-2}+U_{r-1}$ -----(2)

(1)-(2): $\displaystyle U_{r}=S_{r}-S_{r-1}$
Yeah. I see. But how on Earth would you come up with that first? I mean, say you wanted to come up with this formula for the rth term, and you don't know that it's $\displaystyle S_{r}-S_{r-1}$? What we did now there was to confirm that $\displaystyle U_{r} = S_{r}-S_{r-1}$. There is no way I could have simply thought: Ah, find the difference between the sum of the first r terms and the sum of the first (r-1) terms.

4. Look at how $\displaystyle S_{r}$ is defined.

$\displaystyle S_{r}=U_{1}+U_{2}+U_{3}+...+U_{r-1}+U_{r}$

Then consider
$\displaystyle S_{r-1}=U_{1}+U_{2}+U_{3}+...+U_{r-2}+U_{r-1}$

If I take the difference, then I get a formula for the rth term, $\displaystyle U_{r}=S_{r}-S_{r-1}$ right?

5. Originally Posted by acc100jt
Look at how $\displaystyle S_{r}$ is defined.

$\displaystyle S_{r}=U_{1}+U_{2}+U_{3}+...+U_{r-1}+U_{r}$

Then consider
$\displaystyle S_{r-1}=U_{1}+U_{2}+U_{3}+...+U_{r-2}+U_{r-1}$

If I take the difference, then I get a formula for the rth term, $\displaystyle U_{r}=S_{r}-S_{r-1}$ right?
Oh, right.

I didn't see the $\displaystyle U_{r}$ at the end of the $\displaystyle S_{r}$ defintion.

Many thanks, Acc100jt.