Problem 77
Question
If \((1+x)^{15}=C_{0}+C_{1} x+C_{2} x^{2}+\ldots+C_{15} x^{15}\), then the value of \(C_{2}+2 C_{3}+3 C_{4}+\ldots+14 C_{15}\) is (A) 219923 (B) 16789 (C) 219982 (D) none of these
Step-by-Step Solution
Verified Answer
None of these
1Step 1: Understanding the Problem
We need to find the value of the expression \(C_{2}+2 C_{3}+3 C_{4} + \ldots+14 C_{15}\) for the binomial expansion of \((1+x)^{15}\). The terms where \(C_n\) are the binomial coefficients \(\binom{15}{n}\).
2Step 2: Analyzing the Expression
Observe that the expression \(C_{2} + 2C_{3} + 3C_{4} + \ldots + 14C_{15}\) is a weighted sum of the coefficients. We need to consider how to evaluate this sum in a systematic way.
3Step 3: Differentiating the Binomial Expansion
We take the derivative of the binomial expansion and set \(x = 1\). The result will be the sum of terms weighted by their index. Differentiate \((1+x)^{15}\) to get \(15(1+x)^{14}\).
4Step 4: Substitute into the Differential Equation
After differentiating, substitute \(x = 1\) into \(15(1+x)^{14}\), which gives \(15 imes 2^{14}\). This represents the sum of weighted coefficients, where each coefficient is weighted by its power.
5Step 5: Calculating the Numerical Value
Calculate \(15 imes 2^{14}\). First, calculate \(2^{14} = 16384\). Then multiply by 15 to get the final result: \[15 \times 16384 = 245760\] which corresponds to the sum of these weighted coefficients.
Key Concepts
Binomial CoefficientsDifferentiationWeighted SumPower of a Binomial
Binomial Coefficients
In the world of mathematics, binomial coefficients play a crucial role in expanding expressions raised to a power using the Binomial Theorem. When you see a term like \((1+x)^{15}\), the expansion can be written as \(C_0 + C_1x + C_2x^2 + \ldots + C_{15}x^{15}\). Here, each \(C_n\) represents a binomial coefficient, specifically \(\binom{15}{n}\) which is calculated as \(\frac{15!}{n!(15-n)!}\).
These coefficients tell us how many ways we can choose \(n\) elements from a set of 15. For example, \(\binom{15}{2}\) indicates the number of ways to choose 2 items from a set of 15, and similarly for other values of \(n\). They help define the structure of the polynomial when expanded. That's why they are key elements in expressing powers of a binomial in expanded form.
These coefficients tell us how many ways we can choose \(n\) elements from a set of 15. For example, \(\binom{15}{2}\) indicates the number of ways to choose 2 items from a set of 15, and similarly for other values of \(n\). They help define the structure of the polynomial when expanded. That's why they are key elements in expressing powers of a binomial in expanded form.
Differentiation
Differentiation is a fundamental concept in calculus that involves finding the rate at which one quantity changes with respect to another. In this exercise, we see it applied to a binomial expression. By differentiating the expression \((1+x)^{15}\), we obtain \(15(1+x)^{14}\).
Differentiating a polynomial effectively means applying the power rule, where the exponent decreases by one and multiplies the coefficient. This process is necessary to derive the formula used in calculating the weighted sum of the binomial coefficients, as it emphasizes the role of each term in the expression. Differentiation simplifies our task of weighting each coefficient by bringing it out to the front.
Differentiating a polynomial effectively means applying the power rule, where the exponent decreases by one and multiplies the coefficient. This process is necessary to derive the formula used in calculating the weighted sum of the binomial coefficients, as it emphasizes the role of each term in the expression. Differentiation simplifies our task of weighting each coefficient by bringing it out to the front.
Weighted Sum
A weighted sum is a type of summation where different elements contribute differently to the final total based on a set of weights. In this particular problem, we have a sum \(C_2 + 2C_3 + 3C_4 + \ldots + 14C_{15}\), where the binomial coefficients \(C_n\) are multiplied by their respective weights \(2, 3, 4, \ldots, 14\).
To compute this sum systematically, we can use differentiation as a tool and substitute specific values into the resulting polynomial, capturing the essence and the contribution of each coefficient. Each \(C_n\) influences the sum depending on its position \(n\), highlighting the importance of setting appropriate weights that reflect the index numbers.
To compute this sum systematically, we can use differentiation as a tool and substitute specific values into the resulting polynomial, capturing the essence and the contribution of each coefficient. Each \(C_n\) influences the sum depending on its position \(n\), highlighting the importance of setting appropriate weights that reflect the index numbers.
Power of a Binomial
The power of a binomial refers to raising an expression to a certain exponent. In our case, the exercise deals with \((1+x)^{15}\). Here, the power indicates the number of times the multiplication operation is applied between the base \((1+x)\).
This power leads to a polynomial where each term consists of terms like \(C_n x^n\). The Binomial Theorem helps us systematically expand and evaluate these expressions, showing the terms and binomial coefficients. Each term includes a combination of the powers of \(1\) and \(x\), contributing to forming the polynomial's overall structure. Understanding powers in binomials clarifies the effect of raising expressions on the structure and coefficients, which is critical when assessing functions and their expansion.
This power leads to a polynomial where each term consists of terms like \(C_n x^n\). The Binomial Theorem helps us systematically expand and evaluate these expressions, showing the terms and binomial coefficients. Each term includes a combination of the powers of \(1\) and \(x\), contributing to forming the polynomial's overall structure. Understanding powers in binomials clarifies the effect of raising expressions on the structure and coefficients, which is critical when assessing functions and their expansion.
Other exercises in this chapter
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