1. Sequence Problem

If The t3=8 And t8=23 In An Arithmetic Sequence, Find a (Intital Term) And d (Common Difference) .

Thanks In Advance .

2. The third term is 8, the 8th term is 23. So, in the space of 5 terms, we covered a "distance" of 15. Thus, the common difference is 15/5 or 3.

To find the first term, let's work backwards from the third term. Since we're working backwards, instead of adding 3, we'll subtract 3.

Second term = 8 - 3 = 5.

Third term = 5 - 3 = 2.

3. Hello, JennV!

If $\displaystyle t_3=8\text{ and }t_8=23$ in an arithmetic sequence,
find $\displaystyle a$ (first term) and $\displaystyle d$ (common difference).
We're expected to be familiar with the formula: .$\displaystyle t_n \:=\:a + (n-1)d$

. . $\displaystyle \begin{array}{cccc}t_3\:=\:8 & \Rightarrow & a + 2d \:=\:8 & {\color{blue}[1]}\\ t_8 \:=\:23 & \Rightarrow & a + 7d \:=\:23 & {\color{blue}[2]}\end{array}$

Subtract [1] from [2]: .$\displaystyle 5d \:=\:15 \quad\Rightarrow\quad \boxed{d \:=\:3}$

Substitute into [1]: .$\displaystyle a + 2(3)\:=\:8 \quad\Rightarrow\quad \boxed{a \:=\:2}$