Answer:
The first term is 14
The common difference is -2.5
Step-by-step explanation:
we know that
The rule to calculate the an term in an arithmetic sequence is
[tex]a_n=a_1+d(n-1)[/tex]
where
d is the common difference
a_1 is the first term
we have that
The third term of an arithmetic sequence is equal to 9
so
[tex]a_3=9[/tex]
[tex]n=3[/tex]
substitute
[tex]9=a_1+d(3-1)[/tex]
[tex]9=a_1+2d[/tex] ----> equation A
The rule to find the sum of the the first n terms of the arithmetic sequence is equal to
[tex]S=\frac{n}{2} [2a_1+(n-1)d][/tex]
we have
The sum of the first 8 term is 42
so
[tex]S=42[/tex]
[tex]n=8[/tex]
substitute
[tex]42=\frac{8}{2} [2a_1+(8-1)d][/tex]
[tex]42=4[2a_1+7d][/tex]
[tex]10.5=2a_1+7d[/tex] ----> equation B
Solve the system of equations
[tex]9=a_1+2d[/tex] ----> equation A
[tex]10.5=2a_1+7d[/tex] ----> equation B
Solve the system by graphing
Remember that the solution is the intersection point both graphs
using a graphing tool
the solution is (14,-2.5)
see the attached figure
therefore
[tex]a_1=14\\d=-2.5[/tex]
The first term is 14
The common difference is -2.5
