Answer: The required equation for the n-th term is \(a_n=-773+48n.\)
Step-by-step explanation: We are given to find the equation for the nth term of the arithmetic sequence with the fifteenth and sixteenth term as follows :
\(a_{15}=-53,~~~a_{16}=-5.\)
We know that
the nth term of an arithmetic sequence with first term a and common difference d is given by
\(a_n=a+(n-1)d.\)
So, we have
\(a_{15}=-53\\\\\Rightarrow a+(15-1)d=-53\\\\\Rightarrow a+14d=-53~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)\)
and
\(a_{16}=-5\\\\\Rightarrow a+(16-1)d=-5\\\\\Rightarrow a+15d=-5~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(ii)\)
Subtracting equation (i) from equation (ii), we get
\((a+15d)+(a+14d)=-5-(-53)\\\\\Rightarrow d=48.\)
And, from equation (i), we get
\(a+14\times48=-53\\\\\Rightarrow a+672-53\\\\\Rightarrow a=-725.\)
Therefore, the n-th term of the given sequence is
\(a_n=a+(n-1)d\\\\\Rightarrow a_n=-725+(n-1)48\\\\\Rightarrow a_n=-773+48n.\)
Thus, the required equation for the n-th term is \(a_n=-773+48n.\)
