George wants to learn $90$ words in $53$ days, and learns a minimum of $1$ word a day. Show that during a span of consecutive days he will learn $15$ words.
I already have the solution. I just need help understanding how they get to the $b_1 + 15, b_2 + 15$ part.
They got to that, by adding the 15 words he is supposed to have learned in consecutive days, to each consecutive day total.
They then start off assuming each value is distinct. This fails, due to there being more values, than the possible range. Proving by distinctness of the consecutive day totals, that at least one of the latter 53 values, overlaps with one of them. This then implies, that at least one of the first 53 values, is 15 more than another of those 53 values.