A sequence generated by the rule that the kth term is k²+1 for each positive integer k. In this sequence, for any value of n>1, the value of the (n+1)th term less the value of the nth term is
a) 1
b) 3
c) n²
d) 2n+1
e) n²+1
The right answer is d). Could someone explain why? Please.
A sequence generated by the rule that the kth term
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- melguy
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n=2
The sequence for 2nd term is 2²+1 = 5
The sequence for 3rd term is 3²+1 = 10
The sequence for 4th term is 4²+1 = 17 etc
The value of the (n+1)th term (3rd term) minus the value of the nth term (2nd term)
10 - 5 = 5 which is equal to (2 x 2) + 1
The sequence for 2nd term is 2²+1 = 5
The sequence for 3rd term is 3²+1 = 10
The sequence for 4th term is 4²+1 = 17 etc
The value of the (n+1)th term (3rd term) minus the value of the nth term (2nd term)
10 - 5 = 5 which is equal to (2 x 2) + 1
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Maybe the best thing to notice first is that since each term has 1 added to it, in calculating the differences, we can ignore the 1's.sofiasol wrote:A sequence generated by the rule that the kth term is k²+1 for each positive integer k. In this sequence, for any value of n>1, the value of the (n+1)th term less the value of the nth term is
a) 1
b) 3
c) n²
d) 2n+1
e) n²+1
In other words, (y² + 1) - (x² + 1) = y² - x²
So basically what we are working with are the differences between the squares of successive integers.
One way to get to the answer, therefore, is to just plug some consecutive integers into the sequence and then see which of the answer choices we can eliminate.
2² = 4
3² = 9
4² = 16
5² = 25
Since the differences between the squares are all greater than 3, we can eliminate A and B.
Since 2² + 2² ≠3², we can eliminate C.
Let's try D.
2² + 2(2) + 1 = 9 = 3²
3² + 2(3) + 1 = 16 = 4²
D is looking good so far. So let's eliminate E, if possible.
3² + 3² + 1 = 19 ≠4²
So E is out.
The correct answer is D.
Now, there is a much faster way to do this. Vision rules on the GMAT, and so by seeing something, you can burn through this question.
We are looking for the difference between n² + 1 and (n + 1)² + 1 which is the same as
(n + 1)² - n².
You can multiply out the first term to get the following.
(n² + 2n + 1) - n² = 2n + 1
The correct answer is D.
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Hi melguy,
TESTing VALUES is a fantastic approach for this question. However, your work is incomplete. If you use N=2, then TWO answers 'match up' with what you're looking for (Answers D and E). Using N=3 would end in just the one correct answer.
GMAT assassins aren't born, they're made,
Rich
TESTing VALUES is a fantastic approach for this question. However, your work is incomplete. If you use N=2, then TWO answers 'match up' with what you're looking for (Answers D and E). Using N=3 would end in just the one correct answer.
GMAT assassins aren't born, they're made,
Rich
- melguy
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Hi Rich
I agree. I will be careful next time.
(sofiasol: Sorry for the inconvenience. The approach is correct so you can still use it but just remember to test with different numbers if you have two options yielding same answer).
Thanks
I agree. I will be careful next time.
(sofiasol: Sorry for the inconvenience. The approach is correct so you can still use it but just remember to test with different numbers if you have two options yielding same answer).
Thanks
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Let's make it a function instead, such that
f(k) = k² + 1
We're asked to find f(n + 1) - f(n), so we'd get
f(n + 1) = (n + 1)² + 1
and
f(n) = n² + 1
Then simply do (n + 1)² + 1 - (n² + 1).
f(k) = k² + 1
We're asked to find f(n + 1) - f(n), so we'd get
f(n + 1) = (n + 1)² + 1
and
f(n) = n² + 1
Then simply do (n + 1)² + 1 - (n² + 1).