Answer:
The nth for the arithmetic sequence is given by:
\(a_n=a+(n-1)d\) ....[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:
\(a_{10} = 32\) and \(a_{12} =106\)
Using [1] we have;
\(a_{10} = a+9d\)
⇒\(a+9d = 32\) ......[2]
\(a_{12} =a+11d\)
⇒\(a+11d =106\) .......[3]
Subtract equation [2] from [3] we have;
\(a+11d-a-9d = 106-32\)
Simplify:
\(2d = 74\)
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;
\(a_n=-301+(n-1)(37)\)
⇒\(a_n = -301+37n-37 = -338+37n\)
Therefore, an equation for the nth term of the arithmetic sequence is :
\(a_n = -338+37n\)
