What is the sum of the squares of the first n positive integers if the

Question

What is the sum of the squares of the first n positive integers if the sum of the squares of the first n even positive integers is A?

A. A(3/4)
B. A(1/2)
C. A(1/3)
D. A(1/4)
E. A(1/8)

BrentGMATPrepNow · Accepted Answer

We want to determine the value of  1² + 2² + 3² + 4² + . . . + n²

Given: The sum of the squares of the first n EVEN positive integers is A  
In other words: 2² + 4² + 6² + 8² + . . . + (2n)² = A
We can rewrite this as: (2 x 1)² + (2 x 2)² + (2 x 3)² + (2 x 4)² + . . . + (2 x n)² = A
Rewrite as: (2²)(1²) + (2²)(2²) + (2²)(3²) + (2²)(4²) + . . . + (2²)(n²) = A
Evaluate 2² to get: (4)(1²) + (4)(2²) + (4)(3²) + (4)(4²) + . . . + (4)(n²) = A 
Factor to get: 4  = A
Divide both sides by 4 to get:  1² + 2² + 3² + 4² + . . . + n² = A/4

Answer: D

Cheers,
Brent

Abhishek009 · Answer

Let  n = 3

So, Sum of squares of the first 3 even positive integers is  2^2 + 4^2 + 6^2 = 56 = A 
And , Sum of squares of the first 3 consecutive positive integers is  1^2 + 2^2 + 3^2 = 14 = \frac{A}{4}

Thus, Answer must be (D)

CoachNikhilesh · Answer

Say the sum of the squares of the first n positive integers (a.k.a the solution) is x,

1^2 + 2^2 + 3^2 + ....... + n^2 = x

Now, the sum of first n positive integers is A,

2^2 + 4^2 + 6^2 + ....... + (2*n)^2 = A
Take 2^2 (=4) out
4*(1^2 + 2^2 +3^2+ ....... + n^2) = A
Replacing the definition of x from above,
4*x = A
Therefore, x = A/4

Therefore,  D. A(1/4)

GMATinsight · Answer

Let, n = 2

if the sum of the squares of the first n even positive integers is A i.e.  A = 2^2+4^2 = 20

SUm of square of n positive integers  = 1^2+2^2 = 5

5 = (1/4)*20 
 i.e. New Sum  = (1/4)*A

Answer: Option D

ScottTargetTestPrep · Answer

Solution:

If we let n = 3, we see that the sum of the squares of the first 3 positive integers is 1^2 + 2^2 + 3^2 = 1 + 4 + 9 = 14 and the sum of the squares of the first 3 even positive integers is 2^2 + 4^2 + 6^2 = 4 + 16 + 36 = 56.

We see that the sum of the squares of the first 3 positive integers is ¼ the sum of the squares of the first 3 even positive integers. It should be true that the sum of the squares of the first n positive integers is also ¼ the sum of the squares of the first n even positive integers. Since the latter sum is given as A, therefore the former sum is A(¼).

Alternate Solution:

The nth positive even integer is 2n. We are given that:

2^2 + 4^2 + 6^2 + … + (2n)^2 = A

Let’s factor out 2 from each base:

(2*1)^2 + (2*2)^2 + (2*3)^2 + … + (2 * n)^2 = A

(2^2)*(1^2) + (2^2)*(2^2) + (2^2)*(3^2) + … + (2^2)*(n^2) = A

4(1^2) + 4(2^2) + 4(3^2) + … + 4*(n^2) = A

4(1^2 + 2^2 + 3^2 + … + n^2) = A

1^2 + 2^2 + 3^2 + … + n^2 = A/4

Answer: D

Adit_ · Answer

Simply take an example and we have our answer.

1^2+2^2+3^2 = 14.
2^2+4^2+6^2 = 56.

14 = 56*1/4

Thus,  option D

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