Answer:
The nth for the arithmetic sequence is given by:
[tex]a_n=a+(n-1)d[/tex] ....[1]
where,
a is the first term
d is the common difference of two consecutive terms.
n is the number of terms.
As per the statement:
[tex]a_{10} = 32[/tex] and [tex]a_{12} =106[/tex]
Using [1] we have;
[tex]a_{10} = a+9d[/tex]
⇒[tex]a+9d = 32[/tex] ......[2]
[tex]a_{12} =a+11d[/tex]
⇒[tex]a+11d =106[/tex] .......[3]
Subtract equation [2] from [3] we have;
[tex]a+11d-a-9d = 106-32[/tex]
Simplify:
[tex]2d = 74[/tex]
Divide both sides by 2 we get;
d = 37
Substitute the value of d in [2] we have;
a+9(37) = 32
a+333 = 32
Subtract 333 from both sides we have;
a = -301
Then substitute the value a and d in [1] we have;
[tex]a_n=-301+(n-1)(37)[/tex]
⇒[tex]a_n = -301+37n-37 = -338+37n[/tex]
Therefore, an equation for the nth term of the arithmetic sequence is :
[tex]a_n = -338+37n[/tex]
