What is the sum of all the integers from −25 to 30 inclusive?

Everything from −25 up to +25 cancels in pairs and contributes nothing. That leaves only 26 + 27 + 28 + 29 + 30 = 140. The long route agrees: there are 56 terms, so the sum is $56\times (-25+30)÷2=56\times 2.5=140$. Starting the leftover run at 25 instead of 26 — forgetting that 25 has already been cancelled by −25 — gives the wrong total 165.