Okay, so I need to simplify this big fraction. The numerator is a product of terms like (2^4 + 1/4), (4^4 + 1/4), up to [(2n)^4 + 1/4]. The denominator is similar but with odd numbers: (1^4 + 1/4), (3^4 + 1/4), up to [(2n - 1)^4 + 1/4]. 

First, let me write out the general term in both numerator and denominator. For the numerator, each term is ( (2k)^4 + 1/4 ), where k goes from 1 to n. For the denominator, each term is ( (2k - 1)^4 + 1/4 ), also for k from 1 to n. 

So the entire expression is the product from k=1 to n of [ (2k)^4 + 1/4 ] divided by the product from k=1 to n of [ (2k - 1)^4 + 1/4 ]. 

Hmm, maybe I can factor each term in the numerator and denominator to see if there's a telescoping product or some cancellation. Let me focus on factoring (2k)^4 + 1/4. 

Wait, (2k)^4 is (16k^4). So the term is 16k^4 + 1/4. Similarly, the denominator term is (2k - 1)^4 + 1/4, which is 16(k - 1/2)^4 + 1/4. Maybe these can be factored using some identity?

I remember that expressions of the form a^4 + b^4 can sometimes be factored into quadratics. Maybe using the Sophie Germain identity? Let me recall: a^4 + 4b^4 = (a^2 + 2ab + 2b^2)(a^2 - 2ab + 2b^2). 

Wait, in this case, we have 16k^4 + 1/4. Let me see: 16k^4 is (2k)^4, so 16k^4 + 1/4 = (2k)^4 + (1/2)^4. Hmm, not sure if that directly applies. Wait, the Sophie Germain identity is a^4 + 4b^4. Let's check:

If we set a = 2k and b = 1/2, then a^4 + 4b^4 = (2k)^4 + 4*(1/2)^4 = 16k^4 + 4*(1/16) = 16k^4 + 1/4. That's exactly our term! So 16k^4 + 1/4 factors as ( (2k)^2 + 2*(2k)*(1/2) + 2*(1/2)^2 )( (2k)^2 - 2*(2k)*(1/2) + 2*(1/2)^2 ). Let me compute that.

First factor: (4k^2 + 2k + 0.5)
Second factor: (4k^2 - 2k + 0.5)

So, 16k^4 + 1/4 = (4k^2 + 2k + 0.5)(4k^2 - 2k + 0.5). Let me check that:

Multiply (4k^2 + 2k + 0.5)(4k^2 - 2k + 0.5):

First terms: 4k^2 * 4k^2 = 16k^4
Outer: 4k^2 * (-2k) = -8k^3
Inner: 2k * 4k^2 = 8k^3
Then 2k*(-2k) = -4k^2
Then 0.5*4k^2 = 2k^2
Then 0.5*(-2k) = -k
Then 0.5*0.5 = 0.25

Adding up all terms:

16k^4 -8k^3 +8k^3 -4k^2 +2k^2 -k +0.25 = 16k^4 + (-8k^3 +8k^3) + (-4k^2 +2k^2) -k +0.25 = 16k^4 -2k^2 -k +0.25. Wait, that's not 16k^4 +1/4. There must be a mistake in the calculation.

Wait, maybe I messed up the multiplication. Let me do it step by step:

(4k^2 + 2k + 0.5)(4k^2 - 2k + 0.5)

Multiply term by term:

First, 4k^2 * 4k^2 = 16k^4

Then, 4k^2 * (-2k) = -8k^3

Then, 4k^2 * 0.5 = 2k^2

Next, 2k * 4k^2 = 8k^3

Then, 2k * (-2k) = -4k^2

Then, 2k * 0.5 = k

Then, 0.5 * 4k^2 = 2k^2

Then, 0.5 * (-2k) = -k

Then, 0.5 * 0.5 = 0.25

Now, add all these terms:

16k^4 -8k^3 +2k^2 +8k^3 -4k^2 +k +2k^2 -k +0.25

Combine like terms:

16k^4 + (-8k^3 +8k^3) + (2k^2 -4k^2 +2k^2) + (k -k) +0.25

Simplify:

16k^4 + 0k^3 + 0k^2 + 0k +0.25 = 16k^4 +0.25. Which is 16k^4 +1/4. Perfect! So the factorization is correct.

Therefore, each term in the numerator, which is 16k^4 +1/4, factors into (4k^2 + 2k + 0.5)(4k^2 - 2k + 0.5). Let me write that as (4k^2 + 2k + 1/2)(4k^2 - 2k + 1/2). Similarly, maybe the denominator terms can be factored in the same way?

Wait, the denominator terms are (2k -1)^4 +1/4. Let me check if that can be factored similarly. Let's set m = 2k -1. Then the term is m^4 +1/4. Using the same Sophie Germain identity, m^4 +4*(1/2)^4 = m^4 +1/4. So, m^4 +1/4 factors as (m^2 + 2m*(1/2) + 2*(1/2)^2)(m^2 -2m*(1/2) + 2*(1/2)^2). Let's compute that.

First factor: m^2 + m + 0.5

Second factor: m^2 - m + 0.5

So substituting back m = 2k -1, we get:

First factor: (2k -1)^2 + (2k -1) + 0.5

Second factor: (2k -1)^2 - (2k -1) + 0.5

Let's compute each factor.

First factor:

(2k -1)^2 = 4k^2 -4k +1

Add (2k -1): 4k^2 -4k +1 +2k -1 = 4k^2 -2k

Add 0.5: 4k^2 -2k +0.5

Second factor:

(2k -1)^2 =4k^2 -4k +1

Subtract (2k -1): 4k^2 -4k +1 -2k +1 =4k^2 -6k +2

Add 0.5:4k^2 -6k +2.5

Wait, but 4k^2 -6k +2.5 can be written as 4k^2 -6k + 5/2. Hmm, that's different from the numerator factors. Let me check my calculations again.

Wait, maybe I made a mistake when expanding. Let's do it step by step.

First factor: (2k -1)^2 + (2k -1) + 0.5

First term: (2k -1)^2 =4k^2 -4k +1

Second term: (2k -1) =2k -1

Third term: 0.5

Adding all three: 4k^2 -4k +1 +2k -1 +0.5 =4k^2 -2k +0.5. That's correct.

Second factor: (2k -1)^2 - (2k -1) +0.5

First term: 4k^2 -4k +1

Second term: - (2k -1) = -2k +1

Third term: +0.5

Adding all three:4k^2 -4k +1 -2k +1 +0.5 =4k^2 -6k +2.5. Which is 4k^2 -6k +5/2. Hmm, that seems different from the numerator's factors. 

Wait, but in the numerator, we had (4k^2 +2k +1/2)(4k^2 -2k +1/2). So, the denominator factors are (4k^2 -2k +0.5)(4k^2 -6k +2.5). Not sure if these can be related.

But maybe if we index the denominator differently? Let's see. Let me think about the denominator's product. The denominator is product over k=1 to n of [(2k -1)^4 +1/4], which factors into product over k=1 to n of [ (4k^2 -2k +0.5)(4k^2 -6k +2.5) ].

Wait, but perhaps there's a shift in the index for the denominator factors. Let me consider substituting m = k -1 in one of the factors. Let's see.

Take the first factor in the denominator: 4k^2 -2k +0.5. Let me see if this can be rewritten as 4(k - something)^2 + something. Let's complete the square.

4k^2 -2k +0.5 =4(k^2 - (1/2)k) +0.5

Complete the square inside the parentheses:

k^2 - (1/2)k =k^2 - (1/2)k + (1/16) - (1/16) = (k - 1/4)^2 - 1/16

Multiply by 4: 4(k -1/4)^2 - 4*(1/16) =4(k -1/4)^2 -1/4

Add 0.5: 4(k -1/4)^2 -1/4 +0.5 =4(k -1/4)^2 +1/4

Hmm, that's interesting. So 4k^2 -2k +0.5 =4(k -1/4)^2 +1/4. Similarly, the second factor in the denominator:4k^2 -6k +2.5. Let's try completing the square.

4k^2 -6k +2.5 =4(k^2 - (3/2)k) +2.5

Complete the square:

k^2 - (3/2)k =k^2 - (3/2)k + (9/16) - (9/16) = (k - 3/4)^2 -9/16

Multiply by 4:4(k - 3/4)^2 -4*(9/16)=4(k -3/4)^2 -9/4

Add 2.5:4(k -3/4)^2 -9/4 +2.5=4(k -3/4)^2 -2.25 +2.5=4(k -3/4)^2 +0.25=4(k -3/4)^2 +1/4

So both factors in the denominator can be written as 4(k - a)^2 +1/4 where a is 1/4 and 3/4 respectively. 

Similarly, the numerator factors are 4k^2 +2k +0.5 and 4k^2 -2k +0.5. Let's check if those can be rewritten as squares. 

Take the first numerator factor:4k^2 +2k +0.5. Let's complete the square.

4k^2 +2k +0.5=4(k^2 + (1/2)k) +0.5

Complete the square:

k^2 + (1/2)k =k^2 + (1/2)k + (1/16) - (1/16)= (k +1/4)^2 -1/16

Multiply by 4:4(k +1/4)^2 -4*(1/16)=4(k +1/4)^2 -1/4

Add 0.5:4(k +1/4)^2 -1/4 +0.5=4(k +1/4)^2 +1/4

Similarly, the second numerator factor:4k^2 -2k +0.5. Let's complete the square.

4k^2 -2k +0.5=4(k^2 - (1/2)k) +0.5

Complete the square:

k^2 - (1/2)k =k^2 - (1/2)k + (1/16) - (1/16)= (k -1/4)^2 -1/16

Multiply by 4:4(k -1/4)^2 -4*(1/16)=4(k -1/4)^2 -1/4

Add 0.5:4(k -1/4)^2 -1/4 +0.5=4(k -1/4)^2 +1/4

So the numerator factors are 4(k +1/4)^2 +1/4 and 4(k -1/4)^2 +1/4. 

Wait, this is interesting. So both numerator and denominator factors are of the form 4(k ± a)^2 +1/4. Let me write them:

Numerator factors:
1. 4(k +1/4)^2 +1/4
2. 4(k -1/4)^2 +1/4

Denominator factors:
1. 4(k -1/4)^2 +1/4
2. 4(k -3/4)^2 +1/4

So, when we take the ratio of numerator to denominator, each term is [4(k +1/4)^2 +1/4] / [4(k -1/4)^2 +1/4] * [4(k -1/4)^2 +1/4] / [4(k -3/4)^2 +1/4]

Wait, but that seems like the second term in numerator cancels with the first term in denominator. Let me see.

Wait, actually, the entire numerator is the product over k=1 to n of [4(k +1/4)^2 +1/4] * [4(k -1/4)^2 +1/4]

Denominator is the product over k=1 to n of [4(k -1/4)^2 +1/4] * [4(k -3/4)^2 +1/4]

So, when you take the ratio, most terms will cancel. Let's see:

For each k from 1 to n, the numerator has factors:

Term1: 4(k +1/4)^2 +1/4

Term2: 4(k -1/4)^2 +1/4

Denominator has factors:

TermA: 4(k -1/4)^2 +1/4

TermB:4(k -3/4)^2 +1/4

So when we take the ratio for each k, we get [Term1 * Term2] / [TermA * TermB] = [Term1 / TermB] * [Term2 / TermA]

But Term2 in numerator is TermA in denominator for k-1? Wait, let's see.

Wait, maybe let's substitute m = k -1 in one of the denominator terms.

Wait, TermA in denominator is 4(k -1/4)^2 +1/4. If we let m = k -1, then TermA becomes 4(m + 3/4)^2 +1/4. Hmm, not sure. Alternatively, maybe shift the index.

Wait, let's consider the denominator factors. The first factor in the denominator is 4(k -1/4)^2 +1/4, which is the same as the second factor in the numerator for k-1. Wait, let's check:

If in the numerator, for term k-1, we have 4((k-1) +1/4)^2 +1/4 =4(k - 3/4)^2 +1/4. Which is exactly the first denominator factor. So TermA in denominator is equal to the second numerator factor for k-1. Similarly, the second denominator factor is 4(k -3/4)^2 +1/4, which is equal to the second numerator factor for k-2?

Wait, maybe this is getting too convoluted. Let me try writing out the terms for small n to see a pattern.

Let's take n=1:

Numerator: (2^4 +1/4) =16 +0.25=16.25

Denominator: (1^4 +1/4)=1 +0.25=1.25

Ratio:16.25 /1.25=13=1 +12=13. Hmm, 13. Wait, 13 is (4*1 + 1) * something? Not sure.

Wait, but maybe better to factor the terms:

For n=1:

Numerator: [16*1^4 +1/4] = [4*1^2 +2*1 +0.5][4*1^2 -2*1 +0.5] = (4 +2 +0.5)(4 -2 +0.5)=(6.5)(2.5)=16.25

Denominator: [1^4 +1/4] =1 +0.25=1.25. Which factors as [ (1)^2 + (1)^2 + ... ] Wait, no. Wait, denominator term for k=1 is (1^4 +1/4). Using the same factorization as numerator, but with m=2k-1=1. So [m^4 +1/4] factors into [m^2 + m +0.5][m^2 -m +0.5]. For m=1: [1 +1 +0.5][1 -1 +0.5]= [2.5][0.5]=1.25. 

So, the ratio for n=1 is [6.5 * 2.5] / [2.5 * 0.5] = (16.25)/(1.25)=13. 

Similarly, for n=2:

Numerator: (2^4 +1/4)(4^4 +1/4)= [16 +0.25][256 +0.25]=16.25 *256.25

Denominator: (1^4 +1/4)(3^4 +1/4)= [1 +0.25][81 +0.25]=1.25 *81.25

Compute ratio: (16.25 *256.25)/(1.25 *81.25). Let's compute step by step.

16.25 /1.25=13

256.25 /81.25=256.25 ÷81.25. 81.25*3=243.75, so 256.25 -243.75=12.5. So 3 +12.5/81.25=3 + 0.15384615≈3.1538. But exact value: 256.25 /81.25=25625/8125= divide numerator and denominator by 25: 1025/325= divide by 5: 205/65= divide by 5 again:41/13≈3.1538. So total ratio is 13 *41/13=41. Wait, 13*(41/13)=41. So for n=2, ratio is 41.

Similarly, n=3:

Numerator: (2^4 +1/4)(4^4 +1/4)(6^4 +1/4)=16.25 *256.25 *1296.25

Denominator:(1^4 +1/4)(3^4 +1/4)(5^4 +1/4)=1.25 *81.25 *625.25

Compute ratio:

16.25 /1.25=13

256.25 /81.25=41/13 (from before)

1296.25 /625.25= Let's compute 1296.25 ÷625.25. 625.25*2=1250.5. 1296.25 -1250.5=45.75. So 2 +45.75/625.25. 45.75/625.25= multiply numerator and denominator by 4 to eliminate decimals: 183/2501. Approximately 0.073. So total ratio≈2.073. But exact value:1296.25=1296 +0.25= (5184 +1)/4=5185/4. 625.25=625 +0.25=2501/4. So 5185/4 ÷2501/4=5185/2501. Let's divide 5185 by2501. 2501*2=5002. 5185-5002=183. So 2 +183/2501. So total ratio is 13*(41/13)*(5185/2501)= (13*41*5185)/(13*2501)= (41*5185)/2501. Let's compute 41*5185: 40*5185=207400, 1*5185=5185. Total=207400 +5185=212585. Then 212585 ÷2501. Let's divide 212585 by2501. 2501*85=212585. So 85. Therefore, the ratio is 85.

So for n=1:13, n=2:41, n=3:85. Hmm, seems like the pattern is 4n +9? Wait, 13=4*1 +9=13, 41=4*10 +1=41? Wait, 4*10 +1=41. Wait, 13=4*3 +1, no. Wait, 13,41,85. Let's compute differences:41-13=28, 85-41=44. Not an arithmetic sequence. Ratios:41/13≈3.15, 85/41≈2.07. Hmm, not obvious. Maybe the ratio for general n is (4n +1)^2? Let's check:

For n=1: (4*1 +1)^2=25≠13. No. Wait, 13= (2*1 +3)^2 -4? 5^2 -4=21≠13. Not helpful.

Wait, 13=1*13, 41=1*41, 85=5*17. Hmm, not obvious. Wait, but 13= (2^4 +1)/ (1^4 +1)= (16 +1)/ (1 +1)=17/2=8.5? No. Wait, 13 is 16 +1/4 divided by1 +1/4, but that gives (16.25)/(1.25)=13. So maybe the ratio is (4n +1). Wait, for n=1, 4*1 +1=5≠13. Hmm. Wait, 13= (4*1 +1)^2 - (something). Not sure.

Alternatively, maybe the ratio for n is (4n +1) multiplied by something. Wait, n=1:13= (4*1 +1)*something=5*2.6=13. Not helpful.

Wait, perhaps the general term after cancelling is telescoping. Let's think again about the factors.

Original ratio:

Product_{k=1}^n [ (4k^2 +2k +1/2)(4k^2 -2k +1/2) ] / [ (4k^2 -2k +1/2)(4k^2 -6k +5/2) ]

Wait, but for each k, the (4k^2 -2k +1/2) term cancels between numerator and denominator. So we have:

Product_{k=1}^n [ (4k^2 +2k +1/2) / (4k^2 -6k +5/2) ]

So simplifying, the ratio becomes:

Product_{k=1}^n [ (4k^2 +2k +1/2) / (4k^2 -6k +5/2) ]

Now, perhaps this product telescopes. Let's see if the denominator of one term cancels with the numerator of another term.

Let me write out the terms:

For k=1:

Numerator:4(1)^2 +2(1) +1/2=4 +2 +0.5=6.5

Denominator:4(1)^2 -6(1) +5/2=4 -6 +2.5=0.5

So term1:6.5 /0.5=13

For k=2:

Numerator:4(4) +4 +0.5=16 +4 +0.5=20.5

Denominator:4(4) -12 +2.5=16 -12 +2.5=6.5

Term2:20.5 /6.5≈3.15

For k=3:

Numerator:4(9) +6 +0.5=36 +6 +0.5=42.5

Denominator:4(9) -18 +2.5=36 -18 +2.5=20.5

Term3:42.5 /20.5=2.073...

So the product is 13 *3.15 *2.073...≈13*3.15=40.95*2.073≈84.15. Which is close to 85, which matches our previous calculation for n=3.

But to see if it's telescoping, let's see if the denominator for term k cancels with the numerator for term k-1 or k+1.

Wait, in the denominator of term k:4k^2 -6k +5/2. Let me see if this can be written as the numerator of term (k-1) or term (k+1).

Compute numerator for term (k-1):

4(k-1)^2 +2(k-1) +1/2=4(k^2 -2k +1) +2k -2 +0.5=4k^2 -8k +4 +2k -2 +0.5=4k^2 -6k +2.5

Compare with denominator for term k:4k^2 -6k +5/2=4k^2 -6k +2.5. Wait, exactly the same! So denominator of term k is equal to numerator of term (k-1). Therefore, the product telescopes:

Each denominator cancels with the numerator of the previous term. 

Therefore, the entire product becomes:

[ Numerator of term1 / Denominator of term1 ] * [ Denominator of term1 / Numerator of term2 ] * [ Denominator of term2 / Numerator of term3 ] * ... * [ Denominator of term(n-1) / Numerator of termn ]

Therefore, all intermediate terms cancel, leaving:

Numerator of term1 / Denominator of termn

Wait, let's verify:

First term: (Numerator1 / Denominator1)

Second term: (Numerator2 / Denominator2)

But Denominator1 = Numerator(k=1) = Numerator1. Then Denominator2 = Numerator(k=2). Wait, no. Wait, Denominatork = Numerator(k-1). So Denominator2 = Numerator1. Therefore, the product becomes:

(Numerator1 / Denominator1) * (Numerator2 / Denominator2) * ... * (Numerator n / Denominator n) = (Numerator1 / Denominator1) * (Numerator2 / Numerator1) * (Numerator3 / Numerator2) * ... * (Numerator n / Numerator n-1) * (Numerator n / Denominator n)

Wait, this seems like most terms cancel except the first Numerator and the last Denominator. Wait, let me index it properly.

Suppose we have terms from k=1 to k=n. Each term is (Numeratork / Denominatork). But Denominatork = Numerator(k-1). Therefore, the product becomes:

Product_{k=1}^n [Numeratork / Numerator(k-1)] where Numerator0 is defined as 1 (since for k=1, Denominator1 = Numerator0). 

Therefore, the product telescopes to Numeratern / Numerator0 = Numeratern /1 = Numeratern.

Wait, but Numeratern is the numerator term for k=n:4n^2 +2n +0.5. So the entire product is equal to 4n^2 +2n +0.5. But wait, for n=1, 4(1)^2 +2(1) +0.5=4 +2 +0.5=6.5. But earlier, the ratio for n=1 was 13. Wait, that's not matching. Hmm, maybe my indexing is off.

Wait, let me re-express. The product is:

Product_{k=1}^n [Numeratork / Denominatork] = Product_{k=1}^n [Numeratork / Numerator(k-1)].

But Numerator(k-1) for k=1 is Numerator0. Let's define Numerator0 as 1 (since there's no term for k=0). So the product becomes:

Product_{k=1}^n [Numeratork / Numerator(k-1)] = Numeratern / Numerator0 = Numeratern /1 = Numeratern.

But for n=1, Numeratern is Numerator1 =4*1^2 +2*1 +0.5=6.5. But the actual ratio for n=1 was 13. So there's a discrepancy here. Wait, perhaps I missed a factor. Because when we factor the original expression, we had a cancellation of (4k^2 -2k +0.5) between numerator of k and denominator of k-1. But in reality, the ratio after cancellation is:

[N1 * N2 * ... * Nn] / [D1 * D2 * ... * Dn] = [N1 / D1] * [N2 / D2] * ... * [Nn / Dn]

But since Dk = N(k-1), the product becomes:

[N1 / D1] * [N2 / N1] * [N3 / N2] * ... * [Nn / N(n-1)] = Nn / D1

But D1 = Denominator1 =4*1^2 -6*1 +5/2=4 -6 +2.5=0.5. So the product is Nn / D1 = (4n^2 +2n +0.5)/0.5=8n^2 +4n +1.

Wait, let me check this. For n=1:

Numerator1=6.5, Denominator1=0.5. So 6.5 /0.5=13. Which matches. For n=2:

Numerator2=4*4 +4 +0.5=16 +4 +0.5=20.5. Denominator1=0.5. So 20.5 /0.5=41. Which matches. For n=3:

Numerator3=4*9 +6 +0.5=36 +6 +0.5=42.5. 42.5 /0.5=85. Which matches. So yes, the ratio simplifies to (4n^2 +2n +0.5)/0.5=8n^2 +4n +1. Therefore, the simplified expression is 8n^2 +4n +1.

Wait, but let me confirm with the telescoping product. If the product telescopes to Numeratern / Denominator1, then Numeratern is4n^2 +2n +0.5, and Denominator1 is0.5. So 4n^2 +2n +0.5 divided by0.5 is (4n^2 +2n +0.5)*2=8n^2 +4n +1. So the simplified expression is8n^2 +4n +1. Therefore, the answer is8n² +4n +1. 

Let me verify with n=1:8 +4 +1=13. Correct. n=2:32 +8 +1=41. Correct. n=3:72 +12 +1=85. Correct. So this formula works. Therefore, the simplified expression is8n² +4n +1.

So the final answer is $\boxed{8n^2 + 4n + 1}$.
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