Proof of the arithmetic sequences general formula (not using the 1st term) to find the n-th

Question

Can anyone explain on how come the second form of the formula is valid? Is there anyway to prove its validity without plugging in the numbers?

Formula $1$ : $a_n=a_1+(n-1)d$

Formula $2$ : $a_n=a_m+(n-m)d$

Answer 1

You have $$a_n=a_1+(n-1)d=a_1+nd-d\quad \&\quad a_m=a_1+(m-1)d=a_1+md-d$$ so, by subtracting, we have $$a_n-a_m=nd-md=(n-m)d$$ which is the same as your second formula.