I am given the common difference/ratio and tn = 48 and Sn = 33, I need to find the first term but I am not given any terms how do I do this?
Hello, Cryptnar!
I've made some progress . . .
$\displaystyle \text{I am given the common difference/ratio and }t_n = 48\text{ and }S_n = 33.$
$\displaystyle \text{I need to find the first term, but I am not given any terms. \; How do I do this?}$
Suppose we have a Geometric Progression.
We are given: .$\displaystyle \begin{Bmatrix}t_n &=& ar^{n-1} &=& 48 & [1] \\ \\[-3mm] S_n &=& a\dfrac{1-r^n}{1-r} &=& 33 & [2] \end{Bmatrix}$
From [2]: .$\displaystyle a(1-r^n) \:=\:33(1-r) \quad\Rightarrow\quad a - ar^n \:=\:33 - 33r\;[3] $
Multiply [1] by $\displaystyle r\!:\;ar^n \:=\:48r$
Substitute into [3]: .$\displaystyle a - 48r \:=\:33 - 33r$
. . Therefore: .$\displaystyle a \:=\:33 + 15r\;\hdots\;\text{ and we are given }r.$