We need to find the sum of the first fifty natural numbers from one to fifty. This is an arithmetic series where the first term is one, the last term is fifty, and we have fifty terms total.
To find the sum of an arithmetic series, we use the formula S n equals n over two times the sum of the first term and the last term. Here n is the number of terms, a one is the first term, and a n is the last term.
Now let's substitute our values into the formula. We have S fifty equals fifty over two times one plus fifty. First, we simplify fifty over two to get twenty-five. Then we add one plus fifty to get fifty-one. Finally, we multiply twenty-five times fifty-one to get one thousand two hundred seventy-five.
We can verify this using Gauss's pairing method. We pair the first and last numbers: one plus fifty equals fifty-one. Then two plus forty-nine equals fifty-one. Each pair sums to fifty-one, and we have twenty-five pairs total. So twenty-five times fifty-one equals one thousand two hundred seventy-five. Therefore, the sum of integers from one to fifty is one thousand two hundred seventy-five.
To summarize what we learned: We identified this as an arithmetic series and used the formula S n equals n over two times the sum of first and last terms. We substituted our values and calculated the result as one thousand two hundred seventy-five. We also verified this using Gauss's pairing method.