Question

Find an explicit formula for the arithmetic sequence 170, 85, 0, −85, ... D(n)=

Answer 1

Numbers that are presented in sequences are called progressions. There can be three types of this: arithmetic progression, geometric progression and harmonic progression. Let's focus on the arithmetic progression.

Arithmetic progression are numbers in the sequence that has a common difference, denoted as d. One way to find this is to subtract adjacent numbers within the sequence.
85 - 170 = -85
0 - 85 = -85
-85 - 0 = -85
So, there is a pattern that the common difference is -85. Now, derived formulas are already set conveniently for substitution. For arithmetic progression, the formula is 

An = A1 + d(n-1)
where
An is the nth term of the sequence
A1 is the 1st term of the sequence
n is the total number of terms in the sequence

Hence, for this particular sequence, A1 = 170. Substituting,

An = 170 + (-85)(n - 1)
An = 170 - 85(n-1)

Simplifying further,
An = 170 - 85n + 85
An = 255 - 85n