Find the Largest possible value of the sum of the arithmetic sequence 85, 78, 71,...

Question

Obviously $a=85$ and $d=-7$ but I cannot figure out how to calculate the maximum value given this?

Answer 1

sum increases while the terms are positive, so compute the number of terms to include the last positive term. Then compute the actual sum.

Answer 2

Hint: Your common difference is negative. Therefore, the largest sum will be the sum of all positive terms. Recall that in an arithmetic sequence with the initial term $u_1$ and common difference $d$, the $n^{th}$ term is found by

$$u_n = u_1+(n-1)d$$

Try to find the greatest value of $n$ such that

$$u_n > 0 \implies u_1+(n-1)d > 0$$

so that you can cover all the positive terms, followed by plugging the value of $n$ to find that $u_n$. Finally sum up all the terms from the initial term to the $n^{th}$ term. You can use

$$S_n = \frac{n(u_1+u_n)}{2}$$