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

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Joined: 02 Sep 2009
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13 May 2020, 06:50
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45% (medium)

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66% (01:40) correct 34% (01:57) wrong based on 44 sessions

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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)

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13 May 2020, 07:56
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Bunuel wrote:
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)

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[1² + 2² + 3² + 4² + . . . + n²] = A
Divide both sides by 4 to get: 1² + 2² + 3² + 4² + . . . + n² = A/4

Cheers,
Brent
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13 May 2020, 08:08
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Bunuel wrote:
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)

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 $$1^2 + 2^2 + 3^2 = 14 = \frac{A}{4}$$

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Re: What is the sum of the squares of the first n positive integers if the  [#permalink]

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13 May 2020, 18:38
Bunuel wrote:
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)

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)
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What is the sum of the squares of the first n positive integers if the  [#permalink]

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13 May 2020, 18:56
Bunuel wrote:
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)

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 $$= (1/4)*A$$

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Re: What is the sum of the squares of the first n positive integers if the  [#permalink]

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16 May 2020, 08:28
Bunuel wrote:
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)

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

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Re: What is the sum of the squares of the first n positive integers if the   [#permalink] 16 May 2020, 08:28