If a sequence of consecutive integers of increasing value has a sum of 63 and a first term of 6, how many integers are in the sequence?
Answer Choices
Correct answer marked belowSee the full step-by-step explanation
You can see the correct answer above. Sign in for free to unlock the complete worked solution.
Track your performance and improve
Get detailed analytics, unlock full explanations, and move up difficulty tiers as you practice.
Unlock the full explanation
Create a free account to reveal the correct answer, see the step-by-step explanation, and start tracking your GMAT progress.
Consecutive integers starting from 6... just sum them up until you hit 63. 6+7+8+9+10+11+12+13 = 76, so that's too many. I got 7 terms? Let me check: 6 to 12 inclusive is 7 numbers: 6+7=13, +8=21, +9=30, +10=40, +11=51, +12=63. Yes, 7. But the answer choices have 7 as E. Is that right?
Yep, 7 is correct. You can also use the formula for sum of consecutive integers: n/2 * (first + last) = 63. With first=6 and n terms, last = 6 + n - 1 = n+5. So n/2 * (6 + n+5) = n/2 * (n+11) = 63 => n(n+11)=126 => n^2+11n-126=0 => (n+18)(n-7)=0 => n=7. Works perfectly.
I initially thought it was 8 because 6+7+...+13=76, but then realized 63 is less. So 7 is the only one that works. The answer is E. But I'm curious: is there a faster way than trial and error?
You can use average: sum = average * number of terms. For consecutive integers, average = (first+last)/2. Since last = first + n - 1, average = (2*first + n -1)/2. Set equal to 63/n and solve. But honestly, for this level, plugging in answer choices is quickest.
This one was pretty straightforward. The sequence is 6,7,8,9,10,11,12. Sum is 63. Count is 7. Answer E. I think this is more like 500-level, not 555-605. Maybe the difficulty is because of the wording?
I agree, it felt easy. But sometimes the consecutive integer sum questions trip people up if they don't start at 1. Here starting at 6 is a small twist.
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.