Simple Sequence
Consider the sequence where there are copies of the whole number Find the sum of the first 1000 terms of this sequence.
The answer is 29820.
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1 solution
In this series, the n(n+1)/2 th term is n. After n(n+1)/2 th term comes n+1, n+1, n+1....(n+1 times). Therefore to find the 1000th term, we have
n(n+1)/2 < 1000
From here, n = 44
(44x45)/2 = 990
Therefore the next 10 terms are 45s.
Hence sum of the first 990 terms = n(n+1)(2n+1)/6
Putting n = 44, we get 29370
The next 10 terms add upto 45x10 = 450.
Thus, Sum of first 1000 terms = 29370 + 450 = 29820