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# If the sequence a(n) is defined as a(n) = n^2+n+n+3, then which of the

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If the sequence a(n) is defined as a(n) = n^2+n+n+3, then which of the [#permalink]

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03 Mar 2015, 06:18
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If the sequence a(n) is defined as $$a_n = n^2+n+\sqrt{n+3}$$, then which of the following values of n represents the first term such that a(n) > 500?

A. 13
B. 22
C. 33
D. 46
E. 78

Kudos for a correct solution.
[Reveal] Spoiler: OA

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Re: If the sequence a(n) is defined as a(n) = n^2+n+n+3, then which of the [#permalink]

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03 Mar 2015, 07:35
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IMO it should be B ie 22

(22)^2 + 22 + (22+3)^0.5 = 511
that is greater than 500

Kudos if its a correct answer and still i need to know a fast and smart approach
thanks
celestial

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Re: If the sequence a(n) is defined as a(n) = n^2+n+n+3, then which of the [#permalink]

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03 Mar 2015, 08:01
Celestial09 wrote:
IMO it should be B ie 22

(22)^2 + 22 + (22+3)^0.5 = 511
that is greater than 500

Kudos if its a correct answer and still i need to know a fast and smart approach
thanks
celestial

hi,
if u look at the eq.. $$n^2$$+n+$$\sqrt{n+3}$$= n(n+1)+$$\sqrt{n+3}$$..
so n(n+1) should be close to 500... and 500 is 20*25... so the only value available is 22...
ans B
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If the sequence a(n) is defined as a(n) = n^2+n+n+3, then which of the [#permalink]

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03 Mar 2015, 08:23
Substitute n+3 they are all perfect squares $$\sqrt{(n+3)}$$=+- 4,5,6,7,9
then it matter of adding the squares

A. 13 >> 186
B. 22 >>511
C. 33 >>1089+33+6
D. 46
E. 78

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Re: If the sequence a(n) is defined as a(n) = n^2+n+n+3, then which of the [#permalink]

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07 Mar 2015, 16:58
Hi All,

The wording of this prompt provides a HUGE hint as to how you can solve it quickly. While the individual calculation in the prompt might "look" scary, you can actually ignore most of it. Here's why: the prompt asks for the FIRST value that will lead to a result that is greater than 500. Since the answers are NUMBERS, we know that one of the MUST be the first value that fits this description.

We can TEST THE ANSWERS and use the "spread" (and a bit of estimation) to get to the solution.

First, a quick look at Answer A (since 13^2 isn't a terrible calculation)

IF...N = 13
13^2 + 13 = 169 + 13 + a little more = 182 + a little more. This is NOT greater than 500

For the sake of estimation, let's use N = 20
20^2 + 20 = 400 + 20 + a little more at the end + whatever else isn't there because I rounded down from 22 to 20....
This might be greater than 500, but I'm not going to spend time trying to prove it just yet.....

For estimation purposes, let's use N = 30
30^2 = 900, which is WAY TOO BIG, so N = 33 would be even BIGGER. This can't be the answer.

The remaining answers would be even bigger than C, so they're both out. The only answer that makes any sense is B.

[Reveal] Spoiler:
B

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Math Expert
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Re: If the sequence a(n) is defined as a(n) = n^2+n+n+3, then which of the [#permalink]

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08 Mar 2015, 14:36
1
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Expert's post
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This post was
BOOKMARKED
Bunuel wrote:
If the sequence a(n) is defined as $$a_n = n^2+n+\sqrt{n+3}$$, then which of the following values of n represents the first term such that a(n) > 500?

A. 13
B. 22
C. 33
D. 46
E. 78

Kudos for a correct solution.

MAGOOSH OFFICIAL SOLUTION:

For this one, a frontal assault with algebra is likely to be an unproductive disaster. Backsolving is really the way to go. Notice that each of the three terms increases as n increase, so we have reason to believe the value of a(n) increase as n increases. Let’s start with n= 33. We have to do a little arithmetic to figure out that 33^2 = 1089, but then we can calculate that
Attachment:

b_img2.png [ 1.16 KiB | Viewed 1240 times ]

That’s much bigger than 500. Presumably, the very first term larger than 500 would not be that much bigger than 500. Let’s go down one, to (B). Suppose n = 22. Again, we have to do a little arithmetic to calculate that 22^2 = 484, but then we can compute:
Attachment:

b_img3.png [ 1.03 KiB | Viewed 1237 times ]

Aha! This is just over 500, and the answers are spaced out enough, that the next one, (A), will undoubtedly be much smaller. Therefore, answer = B.
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Re: If the sequence a(n) is defined as a(n) = n^2+n+n+3, then which of the [#permalink]

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Re: If the sequence a(n) is defined as a(n) = n^2+n+n+3, then which of the   [#permalink] 21 Aug 2017, 11:20
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