For every positive integer , the th term of a sequence is the total of three consecutive integers starting at . What is the total of terms 1 through 99 of this sequence?
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Took me a minute to parse the wording. So term n is n+(n+1)+(n+2) = 3n+3? Then sum from 1 to 99. I got 15,147 but I'm not 100% sure I added right.
Yeah that's what I did. Sum of 3n from 1 to 99 is 3*(99*100/2)=14,850, then plus 3*99=297 gives 15,147.
Same. I initially thought each term was just n+(n+1)+(n+2) but forgot to multiply the +3 by 99 at the end. Almost picked C.
Is there a faster way than writing out the sequence? I tried to see a pattern but got lost in the arithmetic.
You can group the sum as sum of 3n plus sum of 3. The first part is 3 times the sum of the first 99 integers.
This one seems more like a 600-level question to me. Just plug in n=1 to see the pattern: 1+2+3=6, then n=2: 2+3+4=9, so terms increase by 3. Then use sum of arithmetic sequence.
I agree, but the wording tripped me up at first. Once you see it's arithmetic it's straightforward.
I initially misread and thought it was sum of three consecutive integers starting at n for each term, but then I wasn't sure if the sequence itself is the totals or the integers. The phrasing 'the nth term is the total of' makes it clear though.
What it tests
Your ability to spot and extend arithmetic and geometric sequences and work with recursive definitions.
Common trap
Off-by-one errors in indexing (a₀ vs a₁) or confusing an arithmetic difference with a geometric ratio.