Given that the first term of an arithmetic sequence is - 20 and the 50th term is 120, find the sum of its first 50 terms?

Given that the first term of an arithmetic sequence is - 20 and the 50th term is 120, find the sum of its first 50 terms?

Let the first term of the arithmetic sequence be A1 (for convenience) and the tolerance be d. then the general term an = a1 + (n-1)) × D (n is a natural number)
From 2A6 = A3 + A9 = 50, A6 = 50 / 2 = 25 can be obtained
We also know that A5 × A7 = 616, that is, (a6-D) × (A6 + D) = 616
We can get d = - 3 (sequence is decreasing sequence)
25 = A6 = a1 + (6-1) × (- 3) can get A1 = 40
Then an = 40 + (n-1) (- 3) = 43-3n
When an = 0, 43-3n = 0, n = 14
An0, i.e. 43-3n0, gives n = 15
Then Sn = (a1 + an) × (1 / 2) × n
=[40+(43-3n)]×(1/2)×n
Smax=S14
=(40+43-3×14) ×(1/2) ×14
=41×7
=287