1. Find the arithmetic progression

1. Find the arithmetic progression where t5 = 17 and t12 = 52

2. Find t6 of an arithmetic sequence given that t3 = 5.6 and t12 = 7

3. Find the 7th term of the arithmetic progression whose 5th term is m and whose 11th term is n.

4. Find the value of p so that p + 5, 4p + 3, 8p -2 will form successive terms of an arithmetic progression

2. Hello, nerdzor!

This should get you started . . .

1. Find the arithmetic progression where: $t_5 = 17\text{ and }t_{12} = 52$
$\text{The }n^{th}\text{ term is: }\;t_n \;=\;t_1 + (n-d)d$
. . $\text{where }t_1\text{ is the first term and }d\text{ is the common difference.}$

We have: . $\begin{array}{cccccc}t_5 & = & t_1 + 4d &^ = & 17 & [1] \\ t_{12} &=& t_1 + 11d & = & 52 & [2] \end{array}$

Subtract [1] from [2]: . $7d \:=\:35\quad\Rightarrow\quad\boxed{ d \:=\:5}$

Substitute into [1]: . $t_1 + 4(5) \:=\:17\quad\Rightarrow\quad\boxed{ t_1 \:=\:-3}$

The progression is: . $-3,\:2,\:7,\:12,\:17,\:22,\:27,\:32.\:37.\:42.\:47, \;52,\:\cdots$

4. Find the value of $p$ so that: $p + 5,\;4p + 3,\;8p -2$
are successive terms of an arithmetic progression.
Consecutive terms will have a common difference.

Hence: . $(4p+3) - (p + 5) \;=\;(8p-2) - (4p+3)$

. . And solve for $p\!:\;\;\boxed{p \:=\:3}$

[The terms are: . $8,\:15,\text{ and }22$.]