Question

Find an equation for the nth term of the arithmetic sequence. a15 = -53, a16 = -5

Answer 1

48n-773 is the equation for the nth term

Answer 2

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 $a_n=-773+48n.$