Find the coefficient of the 1996th term of a product

In summary: Are you sure you have the right book?I still see a exponents km and kn that should be km and kn. Are you sure you have the right book?
  • #1
QuietMind
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2

Homework Statement


Let ##\prod_{n=0}^{1996} (1 + nx^{3^n}) = 1 + a_1 x^{k_1} + a_2 x^{k_2} + ... + a_m x^{k^m} ## ,
where ##a_1, a_2, ..., a_m ## are nonzero and ##k_1 < k_2 < ... < k_m ##. Find ##a_{1996}##.
From Art and Craft of Problem Solving, originally from Turkey, 1996

Homework Equations


N/A

The Attempt at a Solution


I think I have a solution, but I'm having difficulty explaining myself. I would like to get an evaluation of this thought process:

I think it is relevant that powers of 3 cannot add up to equal another power of 3 and ##\sum_{k=1}^{n-1} 3^k < 3^n##. This means that when the product is expanded out completely, no two terms will have the same power of x, so no simplification of terms will be possible. Because it is a binomial, this means that the number of terms when expanding out the first k products (ie expand ##\prod_{n=1}^{k} (1 + nx^{3^n}) ##)is equal to twice the number of terms when expanding out the first k-1 products.

We don't need to completely compute out the 1996 factors, as we are essentially being asked to describe the 1996th smallest term (if we compute out all the factors we will have many more than that). These smaller terms would be obtained by taking the ##nx^{k^n}## contribution from some or all of the "smaller" terms and taking the 1 from the "larger" terms.(Because of the properties of the powers of 3 that I stated above, we don't need to consider the factors that contribute too large a power of 3, because the inclusion of their x contribution would instantly push us over the 1996th smallest term. This cutoff happens at n=11) We want to find what sum of powers of 2 would yield 1996, which is ##2^{10} + 2^9 + 2^8 + 2^7 + 2^6 + 2^3 + 2^2 = 1996##.
This tells us that the 10th, 9th, 8th, 7th, 6th, 3rd and 2nd factors contribute their ##nx^{3^n}## term, while the other factors contribute their 1 (so are essentially ignored). The coefficient is then ##10*9*8*7*6*3*2 = 181440##.

Does this seem correct, or can I clarify any steps that are not sufficiently explained? I'm still in the process of learning how to write thorough proofs.

Edit: LaTeX formatting
 
Last edited:
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  • #2
Looks right to me. Well done.
Some of you superscripts/subscripts are a little awry.
 
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  • #3
haruspex said:
Looks right to me. Well done.
Some of you superscripts/subscripts are a little awry.

Thank you! I revised the formatting for anyone who happens to come across this later.
 
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  • #4
QuietMind said:
Thank you! I revised the formatting for anyone who happens to come across this later.
I still see a exponents km and kn that should be km and kn.
 

Related to Find the coefficient of the 1996th term of a product

What is the meaning of "coefficient" in this context?

In mathematics, the coefficient of a term in a product is the number or variable that is multiplied by the other terms in the expression.

How is the 1996th term of a product determined?

The 1996th term of a product can be determined by following the pattern of the product and identifying the term that is in the 1996th position.

What if the product is not in a specific pattern?

If the product is not in a specific pattern, the 1996th term cannot be determined without additional information or context.

Why is finding the coefficient of the 1996th term important?

Finding the coefficient of the 1996th term can help with solving equations, understanding the behavior of a function, and making predictions in mathematical models.

Can the coefficient of the 1996th term be negative?

Yes, the coefficient of the 1996th term can be negative. It depends on the terms and variables in the product and their respective coefficients.

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