The first term of an arithmetic sequence is -3 and the fifteenth term is 53. What is the common difference of the sequence?

so, from the first term, to the fifteen term, there are 14 terms to go through.

that means, if say the common difference is say "d", we have to add "d" 14 times, to get to the 15th term, lemme make a quick table,

[tex]\bf \begin{array}{llll} term&value\\ ----&------\\ 1&-3\\ 2&-3+d\\ 3&-3+d+d\\ 4&-3+d+d+d\\ 5&-3+d+d+d+d\\ 6&-3+d+d+d+d+d\\ 7&-3+d+d+d+d+d+d\\ 8&-3+d+d+d+d+d+d+d\\ 9&-3+d+d+d+d+d+d+d+d\\ 10&-3+d+d+d+d+d+d+d+d+d\\ 11&-3+d+d+d+d+d+d+d+d+d+d\\ 12&-3+d+d+d+d+d+d+d+d+d+d+d\\ 13&-3+d+d+d+d+d+d+d+d+d+d+d+d\\ 14&-3+d+d+d+d+d+d+d+d+d+d+d+d+d\\ 15&-3+d+d+d+d+d+d+d+d+d+d+d+d+d+d\\ &53 \end{array}\\\\ -------------------------------\\\\ -3+14d=53\implies 14d=56\implies d=\cfrac{56}{14}[/tex]

and surely you know how much that is.