Problem 76 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(S_{\mathrm{n}}=1+q+q^{2}+q^{3}+\ldots+q^{\mathrm{n}}\) and \(S_{\mathrm{n}}^{\prime}=1+\left(\frac{q+1}{2}\right)+\left(\frac{q+1}{2}\right)^{2}+\ldots+\left(\frac{q+1}{2}\right)^{n}, q \neq 1\), then \({ }^{\mathrm{n}+1} C_{1}+{ }^{\mathrm{n}+1} C_{2} \cdot S_{1}+{ }^{\mathrm{n}+1} C_{3} \cdot S_{2}+\ldots+{ }^{\mathrm{n}+1} C_{\mathrm{n}+1} \cdot S_{\mathrm{n}}=\) (A) \(2^{n-1} \cdot S_{n}^{\prime}\) (B) \(2^{\mathrm{n}} \cdot S_{\mathrm{n}}\) (C) \(2^{\mathrm{n}+1} \cdot S_{\mathrm{n}}^{\mathrm{n}}\) (D) none of these

Step-by-Step Solution

Answer

The correct answer is (A) \(2^{n-1} \cdot S_n'\).

Step 1: Identify the Series Expressions

We are given two series, \( S_n = 1 + q + q^2 + \ldots + q^n \) and \( S_n' = 1 + \left( \frac{q+1}{2} \right) + \left( \frac{q+1}{2} \right)^2 + \ldots + \left( \frac{q+1}{2} \right)^n \). These are geometric series with common ratios \( q \) and \( \frac{q+1}{2} \) respectively.

Step 2: Find Formula for Geometric Series

The sum of the first \( n+1 \) terms of a geometric series \( 1 + r + r^2 + \ldots + r^n \) is given by \( \frac{r^{n+1} - 1}{r - 1} \), provided \( r eq 1 \). Thus, \( S_n = \frac{q^{n+1} - 1}{q - 1} \) and \( S_n' = \frac{(\frac{q+1}{2})^{n+1} - 1}{(\frac{q+1}{2}) - 1} \).

Step 3: Simplify the Given Expression

The expression is \( \sum_{k=1}^{n+1} { }^{n+1} C_{k} imes S_{k-1} = {}^{n+1} C_{1} + {}^{n+1} C_{2} \cdot S_{1} + \ldots + {}^{n+1} C_{n+1} \cdot S_{n} \). This expression can be simplified using the binomial expansion theorem: \((1+q)^{n+1} = \sum_{k=0}^{n+1} { }^{n+1}C_k \cdot q^k \).

Step 4: Apply Binomial Theorem

Recognize that the term \( (1+q)^{n+1} \) becomes \( (1+q)(1+q)^{n} \). This transformation can be written as \( 1 + \sum_{k=1}^{n+1} { }^{n+1}C_k \cdot q^k \). Relating this to the series, the multiplication and addition in the formula emulate replacing \( q \) with \( \frac{q+1}{2} \).

Step 5: Rewrite the Expression

Using the formula \( S_n' = \frac{(\frac{q+1}{2})^{n+1} - 1}{-\frac{1}{2}} = 2\left((\frac{q+1}{2})^{n+1} - 1\right) \). This shows that the expression \( 2^{n-1} \cdot S_n' = (1+q)^{n+1} - 1 \).

Step 6: Conclusion

After simplification and calculation steps, choice (A) is equivalent to the expression \( 2^{n-1} \cdot S_n' = \sum_{k=1}^{n+1} \binom{n+1}{k} \cdot S_{k-1} \). Thus, option (A) is \( 2^{n-1} \cdot S_n' \).

Key Concepts

Geometric Series

A geometric series is a series of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. For example, in the series \( S_n = 1 + q + q^2 + \ldots + q^n \), the common ratio is \( q \). This type of series is ubiquitous in math due to its neat properties, one of which involves the formula to find the sum of the first \( n+1 \) terms.

The formula for the sum of a geometric series is \( \frac{r^{n+1} - 1}{r - 1} \), where \( r \) is the common ratio and \( r eq 1 \). In our problem, this helped us evaluate both \( S_n \) and \( S_n' \) by substituting the respective common ratios into the formula. For \( S_n \), where the common ratio is \( q \), the result is \( \frac{q^{n+1} - 1}{q - 1} \).

Understanding geometric series is crucial because they form patterns and results that can apply to various fields, such as finance (compound interest), physics (radioactive decay), and computer science (algorithm complexity).

Binomial Theorem

The binomial theorem provides a way to expand expressions of the form \( (a + b)^n \) into a sum involving terms of the products of \( a \) and \( b \). Each term is multiplied by a binomial coefficient denoted as \( \binom{n}{k} \), where \( n \) is the order of the expression, and \( k \) is a specific term.

In the context of our problem, the binomial theorem helps in expanding \( (1 + q)^{n+1} \), which is expressed as \( \sum_{k=0}^{n+1} \binom{n+1}{k} \cdot q^k \). This expansion is key for simplifying complex expressions and for connecting the dots between different mathematical concepts.

The application of the binomial theorem here assists in understanding how combinations of terms can replicate the behavior of substituting one value for another in expressions, making it a versatile tool in problem-solving.

Combinatorics

Combinatorics deals with counting, arrangement, and combination of elements within a set. It is represented in the problem as the binomial coefficients, noted as \( \binom{n+1}{k} \). These coefficients indicate the number of ways to choose \( k \) items from \( n+1 \) without regard to the order.

In our exercise, these coefficients are multiplied with terms from the sum \( S_k \) to form parts of a larger expression. The coefficients embody the core idea of how different choices or combinations contribute numerically to solving the problem.

For instance, if you want to distribute tasks among a group or to create subsets of data, the principles of combinatorics are applied in determining the number of possible solutions. Thus, comprehending these coefficients is vital in mathematical contexts involving arrangements and selections.