Sum of an Arithmetic Sequence
The sum of the fi rst terms of an arithmetic sequence of real numbers is and the sum of the next terms is . Find the largest positive integer for which the sum of the fi rst terms of this sequence is positive.
The answer is 80.
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1 solution
Using the definition of an arithmetic sequence: a(n) = a(1) + (n - 1)(d).
The first equation where the sum of the first 14 terms can be expressed as: 14 a(1) + 91 d = 20 from the formula for arithmetic series.
The second equation can also be expressed as another equation as: 20 a(1) + 470 d = 14 or can further be simplified as 10 a(1) + 235 d = 7.
Solving the system gives a(1) = 239/140 and d = -3/70.
The challenge here is to know what is the maximum k such that the sum of the first k terms of the sequence is positive. We use the arithmetic series where
sum of first k terms = (n/2)(2 a(1) + (n - 1)(d))
Plugging the values solved earlier, we solve the inequality:
(n/2)(239/70 + (n - 1)(-3/70)) > 0
We solve the zeroes of the function above in order to know the bounds, where n = 0 and n = 242/3. Since we are looking for the maximum integer, this corresponds to the max. integer k = floor(max. bound) = floor(242/3) = floor(80 + [2/3]) = 80.