How many terms of the arithmetic progression 3, 8, 13, 18, … must be added to make a total of 570?

Here $a=3$ and $d=5$, so Sₙ = $(n/2)(6+5(n-1))=n(5n+1)/2$. Setting this equal to 570 gives $5n^2+n-1140=0$, whose discriminant $1+22800=22801=151^2$, so $n=(-1+151)/10=15$. (14 and 16 give 497 and 648, which brackets the target and confirms 15 is exact; 10 comes from forgetting the factor 1/2 and solving $n(5n+1)=570$.) Check: S₁₅ = $(15/2)(6+70)=15\times 38=570$.