In the arithmetic sequence, the rule is
[tex]a_n=a_{}+(n-1)d[/tex]
a is the first term and d is the common difference
n is the position of the term
We have the value of the 7th term, then
n = 7, so we can make an equation for it by using the rule above
[tex]\begin{gathered} a_7=a+(7-1)d \\ a_7=a+6d \end{gathered}[/tex]
Substitute a7 by its value -4.2, then
[tex]a+6d=-4.2[/tex]
We will make the same with a23
n = 23, then
[tex]\begin{gathered} a_{23}=a+(23-1)d \\ a_{23}=a+22d \end{gathered}[/tex]
Substitute a23 by its value -7.4
[tex]a+22d=-7.4[/tex]
Now we have a system of equations to solve it to find a and d
a + 6d = -4.2 (1)
a + 22d = -7.4 (2)
Subtract (1) from (2)
(a - a) + (22d - 6d) = (-7.4 - -4.2)
0 + 16d = -3.2
16d = -3.2
Divide both sides by 16 to find d
d = -0.2
To find a substitute d in equation (1) by -0.2
a + 6(-0.2) = -4.2
a - 1.2 = -4.2
Add 1.2 to both sides
a = -3
a) the first term is -3
b) the common difference is -0.2
To find the term 67th substitute n by 67
[tex]\begin{gathered} a_{67}=-3+(67-1)(-0.2) \\ a_{67}=-3+66(-0.2) \\ a_{67}=-3+-13.2 \\ a_{67}=-16.2 \end{gathered}[/tex]
c) the value of the 67th term is -16.2
Let us find the rule of the sequence
[tex]\begin{gathered} a_n=-3+(n-1)(-0.2) \\ a_n=-3+n(-0.2)-1(-0.2) \\ a_n=-3-0.2n+0.2 \\ a_n=-2.8-0.2n \end{gathered}[/tex]
d) The formula of the nth term is
[tex]a_n=-2.8-0.2n[/tex]
