Question: If the average of 18 consecutive odd integers is 534, what is the least number in the set?
Ok so I know how to do this algebraically, you start with your odd variable which we will call X and then the next one will be X + 2, X + 4, etc until you have 18 terms. Then it will be 18X + 306/18 = 534 and then we can solve, but this method seems rather long. Is this the best way to solve this problem?
There are three sum formulas that you should commit to memory:
1. Sum of the first n consecutive integers: n(n+1)/2
2. Sum of the first n consecutive even integers: n(n+1)
3. sum of the first n consecutive odd integers: n^2
So, back to the problem, we have: sum / number of odd integers = [n + (n+2) + (n+4) + ... + (n+34)] / 18 = 534. We can reduce this to 18n + (2 + 4 + 6 + ... + 34) / 18 = 9612 ==> 18n + (2+ 4 + ... + 34) = 173016. We can simplify the sum in the parenthesis using rule 2 from above: 17 x 18 = 306. So we have 18n + 306 / 18 = 534. And now you can solve in the usual way.
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