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# If n is the square of a positive integer, which of the following must

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Math Expert
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If n is the square of a positive integer, which of the following must  [#permalink]

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09 Nov 2015, 21:59
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35% (medium)

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66% (01:17) correct 34% (01:20) wrong based on 268 sessions

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If n is the square of a positive integer, which of the following must be equal to the square of the next positive integer?

A. $$\sqrt{n} + 1$$

B. $$n + 1$$

C. $$n^2 + 1$$

D. $$n + 2\sqrt{n} + 1$$

E. $$n^2 + 2n + 1$$

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Re: If n is the square of a positive integer, which of the following must  [#permalink]

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10 Nov 2015, 00:53
1
Let a= +ve integer such that
n= a^2
(a+1)= next +ve integer
(a+1)^2 = a^2 + 1 + 2a

and a=n^(1/2)
Therefore ,
(a+1)^2=n+1+2*n^(1/2)

Alternatively,
we can take values
let n= 4 =a^2
=> a= 2
(a+1)=3^2 = 9
now checking the choices
A. âˆšn + 1
= 2+1 =3
B. n + 1
= 4+1 =5
C. n^2 + 1
= 16+1 =17
D. n + 2âˆšn + 1
= 4+ 2*2 +1 = 9
E. n^2 + 2n + 1
= 16+ 8 + 1 =25
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Re: If n is the square of a positive integer, which of the following must  [#permalink]

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10 Nov 2015, 10:26
1
If n is the square of a positive integer, which of the following must be equal to the square of the next positive integer?

n = (x)^2 where x is a positive integer
To calculate -
(x+1)^2 = x^2 + 2x + 1
root(n) = x

Ans - n + 2 root(n) + 1
This should be D
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Re: If n is the square of a positive integer, which of the following must  [#permalink]

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08 Oct 2016, 13:59
Bunuel wrote:
If n is the square of a positive integer, which of the following must be equal to the square of the next positive integer?

A. âˆšn + 1
B. n + 1
C. n^2 + 1
D. n + 2âˆšn + 1
E. n^2 + 2n + 1

next positive integer is sqrt(n)+1.
square of this number is [sqrt(n)+1]^2
the result is definitely a number with an "n" not squared not under radical
A,C, E out
so it must be D.

not even needed to square everything...
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Re: If n is the square of a positive integer, which of the following must  [#permalink]

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16 Oct 2016, 13:43
Bunuel wrote:
If n is the square of a positive integer, which of the following must be equal to the square of the next positive integer?

A. âˆšn + 1
B. n + 1
C. n^2 + 1
D. n + 2âˆšn + 1
E. n^2 + 2n + 1

n is +ve integer

n = x^2 thus sqrt n = x , (x+1)^2 = x^2 + 2x+1 , substitute n^2+2sqrtn+1
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Re: If n is the square of a positive integer, which of the following must  [#permalink]

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16 Oct 2016, 22:45
1
Bunuel wrote:
If n is the square of a positive integer, which of the following must be equal to the square of the next positive integer?

A. âˆšn + 1
B. n + 1
C. n^2 + 1
D. n + 2âˆšn + 1
E. n^2 + 2n + 1

Plug in some value and check -

Let n = 1 ( least number )
Square of the next positive integer is 4

Now, check the options : Only (D) and (E) satisfies

Check using one more square number to confirm the answer

Let n = 4 ( least number )
Square of the next positive integer is 9

Now, check the options : Only (D) satisfies

Hence answer must be option (D)

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Re: If n is the square of a positive integer, which of the following must  [#permalink]

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13 Apr 2017, 03:43
1
Option D

$$n = K^2$$; K>0 and Integer

$$(K+1)^2 = K^2 + 2K + 1 = n + 2\sqrt{n} + 1$$
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If n is the square of a positive integer, which of the following must  [#permalink]

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26 May 2017, 08:37
Skywalker18 wrote:
Let a= +ve integer such that
n= a^2
(a+1)= next +ve integer
(a+1)^2 = a^2 + 1 + 2a

and a=n^(1/2)
Therefore ,
(a+1)^2=n+1+2*n^(1/2)

Alternatively,
we can take values
let n= 4 =a^2
=> a= 2
(a+1)=3^2 = 9
now checking the choices
A. âˆšn + 1
= 2+1 =3
B. n + 1
= 4+1 =5
C. n^2 + 1
= 16+1 =17
D. n + 2âˆšn + 1
= 4+ 2*2 +1 = 9
E. n^2 + 2n + 1
= 16+ 8 + 1 =25

This question makes no sense to me . Isn't 5 the next positive integer after 4, and 5^2=25. I definitely did everything you prescribed on my first attempt at this question. Could you elaborate . Answer seems like it should be E to me . It always give you the square of the next positive integer . I am completely perplexed here. Think I'm just being dumb. Thanks !
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Joined: 02 Sep 2009
Posts: 51072
Re: If n is the square of a positive integer, which of the following must  [#permalink]

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26 May 2017, 08:48
2
1
NanaA wrote:
Skywalker18 wrote:
Let a= +ve integer such that
n= a^2
(a+1)= next +ve integer
(a+1)^2 = a^2 + 1 + 2a

and a=n^(1/2)
Therefore ,
(a+1)^2=n+1+2*n^(1/2)

Alternatively,
we can take values
let n= 4 =a^2
=> a= 2
(a+1)=3^2 = 9
now checking the choices
A. âˆšn + 1
= 2+1 =3
B. n + 1
= 4+1 =5
C. n^2 + 1
= 16+1 =17
D. n + 2âˆšn + 1
= 4+ 2*2 +1 = 9
E. n^2 + 2n + 1
= 16+ 8 + 1 =25

This question makes no sense to me . Isn't 6 the next positive integer after 4, and 6^2=36. I definitely did everything you prescribed on my first attempt at this question. Could you elaborate . Think I'm just being dumb. Thanks !

If n is the square of a positive integer, which of the following must be equal to the square of the next positive integer?

A. $$\sqrt{n} + 1$$

B. $$n + 1$$

C. $$n^2 + 1$$

D. $$n + 2\sqrt{n} + 1$$

E. $$n^2 + 2n + 1$$

n is the square of a positive integer. Say that positive integer is x. So, we are given that $$n = x^2$$. Notice that from this it follows that $$x=\sqrt{n}$$

Which of the following must be equal to the square of the next positive integer?

The next positive integer is x + 1. Its square = $$(x+1)^2=x^2+2x+1$$.

Substitute the values of x^2 and x from above to get: $$(x+1)^2=x^2+2x+1=n+2\sqrt{n}+1$$.

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Re: If n is the square of a positive integer, which of the following must  [#permalink]

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17 Jul 2018, 15:45
Bunuel wrote:
If n is the square of a positive integer, which of the following must be equal to the square of the next positive integer?

A. $$\sqrt{n} + 1$$

B. $$n + 1$$

C. $$n^2 + 1$$

D. $$n + 2\sqrt{n} + 1$$

E. $$n^2 + 2n + 1$$

We can let n = k^2 where k is a positive integer, we see that the square of the next positive integer is
(k + 1)^2 = k^2 + 2k + 1. Since n = k^2, so k = âˆšn. Therefore, in terms of n, the square of the next positive integer is n + 2âˆšn + 1.

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Re: If n is the square of a positive integer, which of the following must &nbs [#permalink] 17 Jul 2018, 15:45
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