The problem states that a sequence is defined by the formula $a_n = p + qn$. We are given that the seventh term ($a_7$) is 19 and the fifteenth term ($a_{15}$) is 43. We need to find the values of $p$ and $q$.

1. Problem Description

The problem states that a sequence is defined by the formula $a_n = p + qn$. We are given that the seventh term ($a_7$) is 19 and the fifteenth term ($a_{15}$) is

4

3. We need to find the values of $p$ and $q$.

2. Solution Steps

We are given the following information:

$a_n = p + qn$

$a_7 = 19$

$a_{15} = 43$

Using the formula for $a_n$, we can write equations for $a_7$ and $a_{15}$:

$a_7 = p + 7q = 19$

$a_{15} = p + 15q = 43$

We have a system of two linear equations with two variables, $p$ and $q$:

$p + 7q = 19$

$p + 15q = 43$

We can solve this system of equations using substitution or elimination. Let's use elimination. Subtract the first equation from the second equation:

$(p + 15q) - (p + 7q) = 43 - 19$

$p + 15q - p - 7q = 24$

$8q = 24$

$q = \frac{24}{8}$

$q = 3$

Now, substitute the value of $q$ into the first equation to find $p$:

$p + 7q = 19$

$p + 7(3) = 19$

$p + 21 = 19$

$p = 19 - 21$

$p = -2$

3. Final Answer

$p = -2$

$q = 3$