91Ó°ÊÓ

When faced with a polynomial expression, calculating the coefficient of a specific term can sometimes be tricky. In our polynomial \[ \sum_{k=0}^{1000} (k+1)x^k(1+x)^{1000-k}, \]we aim to find the coefficient of \(x^{50}\). Here, the basic idea is to identify the terms in the polynomial that will contribute to this specific power of \(x\). Each term in the polynomial is of the form \[ (k+1)x^k(1+x)^{1000-k}. \]This contribution can be calculated by using binomial coefficients. Generally, the term contributing to \(x^{50}\) will be given by:

• The product of\( (k+1) \) , which arises from the specific term's multiplier,
• The binomial coefficient \( \binom{1000-k}{50-k} \) that decides the exact arrangement of \(x\) and constant within the term.

Calculate these two components for each term to find its contribution to the overall coefficient of \(x^{50}\). Sum the contributions from qualifying terms, which in this case is \(0 \leq k \leq 50\), for the final result.

A polynomial expression such as \[ (1+x)^{1000} + 2x(1+x)^{999} + 3x^2(1+x)^{998} + \cdots + 1001x^{1000} \]is composed of terms where variables and constants are raised to whole-number exponents.In this particular equation, we see a variable pattern where each term follows a formula\( (k+1)x^k(1+x)^{1000-k} \).

• The term \((1+x)^{1000}\) is the base raised to the highest power, starting the sequence.
• Following this, each subsequent term reduces the power of \( (1+x) \) by 1 and introduces an additional factor \(x\), contributing a higher power of \( x \).

Understanding the structure and pattern of this polynomial helps in systematically analyzing any specific term you are interested in, such as \( x^{50} \) in our example.

The binomial theorem is a powerful tool employed in polynomial expressions, especially involving coefficients. In this problem, combinatorial methods are crucial for determining how different terms contribute to a specific power like \( x^{50} \). The theorem expresses \[ (a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k. \]For our polynomial, combinatorics are used to determine the arrangement coefficients through binomial coefficients \( \binom{1000-k}{50-k} \).

• Identifying valid \(k\) for which the binomial coefficient \( \binom{1000-k}{50-k} \) is non-zero is vital.
• For \( k \) that satisfies \( 0 \leq k \leq 50 \), calculate \( (k+1) \cdot \binom{1000-k}{50-k} \).
• Sum these computations to find the final coefficient.

Using these methods not only simplifies the arithmetic but ensures accuracy when dealing with large exponents and complex expressions. Every combinatorial calculation contributes to understanding the structure of polynomial addition and expansion.