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

Question

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

Skywalker18 · Answer

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)
Answer D

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

keats · Answer

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

mvictor · Answer

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

not even needed to square everything...

yezz · Answer

n is +ve integer

n = x^2 thus sqrt n = x , (x+1)^2 = x^2 + 2x+1 , substitute n^2+2sqrtn+1

Abhishek009 · Answer

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)

RR88 · Answer

Option D

n = K^2 ; K>0 and Integer

(K+1)^2 = K^2 + 2K + 1 = n + 2\sqrt{n} + 1

NanaA · Answer

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 !

Bunuel · Answer

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 .

Answer: D.

JeffTargetTestPrep · Answer

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 = √n. Therefore, in terms of n, the square of the next positive integer is n + 2√n + 1.

Answer: D

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

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

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

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

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards