Question 1209780: Evaluate 21 + 27 + 33 + ... + 261 + 267 + 273 + ... + 6021.
Found 2 solutions by greenestamps, math_tutor2020:
Answer by greenestamps(13200)   (Show Source):
You can put this solution on YOUR website!
21+27+33+...+6021
The common difference is 6, so the formula for the n-th term is of the form
t(n) = 6n+b
t(1) = 21, so the formula is
t(n) = 6n+15
The last term is 6021:
6n+15 = 6021
6n = 6006
n = 1001
The sequence is 1001 terms with first term 21 and last term 6021.
Sum = (number of terms) * (average of all the terms) = (number of terms) * (average of first and last terms)
ANSWER: 3024021
Answer by math_tutor2020(3817)  (Show Source):
You can put this solution on YOUR website!
We can subtract 21 from each term to go from this arithmetic sequence
21,27,33,...,6021
to this arithmetic sequence
0,6,12,...,6000
which are the multiples of 6 in the form 6k, where k ranges from k = 0 to k = 1000
There are 1001 integers in the set {0,1,2,...,1000}
The original arithmetic sequence starts at a1 = 21 and has common difference d = 6.
There are n = 1001 terms being added.
Sn = sum of first n terms of arithmetic sequence
Sn = (n/2)*(2*a1 + d*(n-1))
S1001 = (1001/2)*(2*21 + 6*(1001-1))
S1001 = 3024021
Or,
Sn = (n/2)*(a1 + an)
S1001 = (1001/2)*(a1 + a1001)
S1001 = (1001/2)*(21+6021)
S1001 = 3024021
Therefore,
21+27+33+...+6021 = 3024021
Verification using WolframAlpha
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