# How do I find the common difference if the first term is unknown?

Given the formula: $$A_n = A_1+ (n-1)d$$

I'm trying to look for $$d$$, if given the the second and $$17^{\text{th}}$$ terms, namely $$37$$ and $$82$$.

I can't seem to figure out where to start; if $$A_1$$ were at least given I might have a starting point, but here I'm completely lost now that both are missing.

• Find $A(17)-A(2)$ which should be a computable multiple of $d.$ – coffeemath Jan 27 at 13:51
• How many terms are there from $A_2$ to $A_{17}$? – KM101 Jan 27 at 13:51
• @KM101 Starting at $A(2),$ the common difference $d$ should be added $15$ times to get $A(17.) – coffeemath Jan 27 at 14:00 • @coffeemath For$n$terms, the difference is added$(n-1)$times, I know. This is equivalent to finding$d\$ given the first and sixteenth terms. – KM101 Jan 27 at 14:08
• Then how many d added from second to seventeenth? – coffeemath Jan 27 at 14:12

Hint: $$a_{17} - a_2 = 45 = 16d - d = 15d$$
The $$n$$th term of an arithmetic sequence is
$$a+(n-1)d$$ where $$a$$ is the first term and $$d$$ is the common difference between each term.
$$a+16d=82\tag1$$ $$a+d=37\tag2$$ When we perform $$(1)-(2)$$, what do we get?