Question 1145656
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I'm always curious why nearly all references show the sum of the terms of an arithmetic sequence as<br>
{{{S(n) = (n/2)*(a1+an)}}}<br>
That formula, directly translated into words, says that the sum is half the number of terms, multiplied by the sum of the first and last terms.<br>
To me, a MUCH more meaningful form for the formula is<br>
{{{S(n) = n*((a1+an)/2)}}}<br>
In that form, the formula says the sum is the number of terms, multiplied by the average of the first and last terms.  Since the average of the first and last terms in an arithmetic sequence is the average of ALL the terms, this formula simply states the obvious: a sum is equal to the number of terms, multiplied by the average of the terms.<br>
You can, of course, plug the given information into the formula an solve a quadratic equation to find the answer to the problem.  However, if an algebraic solution is not required, you can find the answer with some rather easy arithmetic.<br>
The sum of the terms of the sequence is 3950; and we know that is the product of  a whole number (the number of terms) and another number which is either a whole number or halfway between two whole numbers.<br>
In this example, with these numbers, by far the most likely case is<br>
3950 = 79*50<br>
So it looks as if it is very likely that there are 50 rows of seats.<br>
Checking that, we see that the 50th row would have 30+49(2) = 30+98 = 128 seats; and that would make the average of the first and last terms (30+128)/2 = 158/2 = 79 -- as required.<br>
So indeed the number of rows of seats is 50.