We need to calculate the sum A equals 1 plus 3 plus 5 plus all the way up to 997 plus 999. This is an arithmetic series of odd numbers. Let's identify the pattern: the first term is 1, and each subsequent term increases by 2, so the common difference is 2.
Now we need to find how many terms are in this series. We use the arithmetic sequence formula: a sub n equals a sub 1 plus n minus 1 times d. Substituting our values: 999 equals 1 plus n minus 1 times 2. Simplifying: 999 equals 1 plus 2n minus 2, which gives us 999 equals 2n minus 1. Adding 1 to both sides: 1000 equals 2n. Therefore, n equals 500. So there are 500 terms in this series.
Now we can calculate the sum using the arithmetic series formula: S equals n over 2 times the sum of the first and last terms. Substituting our values: A equals 500 over 2 times 1 plus 999. This simplifies to A equals 250 times 1000, which gives us our final answer: A equals 250,000.
Let's verify our solution step by step. We identified this as an arithmetic series with first term 1, last term 999, and common difference 2. Using the formula for the nth term, we found there are 500 terms. Then using the sum formula for arithmetic series, we calculated A equals 250,000. This is our final verified answer.
Therefore, the final answer to our problem is A equals 250,000. The sum of all odd numbers from 1 to 999 is 250,000. This problem demonstrates how arithmetic series formulas can efficiently solve complex summation problems that would be very time-consuming to calculate by adding each term individually.