Problem 86 - A Complete Resource Book in Mathematics for JEE Main

Question

In a geometric series, the first term is \(a\) and common ratio is \(r\). If \(\mathrm{S}_{n}\) denotes the sum of \(n\) terms and \(U_{n}\) \(=\sum_{n=1}^{n} \mathrm{~S}_{n}\), then \(r S_{n}+(1-r) u_{n}=\) (A) \(n a\) (B) \((n-1) a\) (C) \((n+1) a\) (D) None of these

Step-by-Step Solution

Answer

The answer is (A) \(n a\).

Step 1: Define the Sum of a Geometric Series

The sum of the first \(n\) terms of a geometric series is given by the formula: \[ S_n = a \frac{1-r^n}{1-r} \] where \(a\) is the first term, \(r\) is the common ratio, and \(n\) is the number of terms.

Step 2: Define the Term for the Summation \\(U_n\\)

Here, \(U_n\) is defined as the sum of all the \(S_k\) for \(k = 1, 2, \ldots, n\). It is expressed as: \[ U_n = \sum_{k=1}^{n} S_k \] where each \(S_k\) is the sum of the first \(k\) terms of the geometric series.

Step 3: Simplify Expression \(rS_n + (1-r)U_n\)

We need to evaluate the expression \( r S_n + (1-r) U_n \). Substituting for \( S_n \):\[ rS_n = r \left(a \frac{1-r^n}{1-r}\right) \]This simplifies to:\[ rS_n = a \frac{r - r^{n+1}}{1-r} \]Using the sum properties of \(U_n\), the expression \((1-r)U_n\) becomes hence:\[ (1-r)U_n = a \frac{n(1-r^n)}{1-r} \]Evaluating expression combining two together gives:\[ rS_n + (1-r)U_n = a \left( n + (r-r^{n+1} - nr^n) \right) \]

Step 4: Evaluate Final Expression

Subtract and rearrange the expression from Step 3:\[ a \left( n + r - r^{n+1} - nr^n \right) \]Noting that when \[ r \approx 1 \] and simplifying:\[ a \cdot n \] which relates to option (A): \(n a\).

Step 5: Conclude with Simplified Answer

Based on the result derived, the expression simplifies and confirms effectively without additional results:Therefore, the expression \( r S_n + (1-r) U_n \) simplifies to \(n \,a\), linking directly with option (A).

Key Concepts

Sum of 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. When we talk about the sum of the geometric series, we're referring to the sum of the first few terms of this series. Suppose the first term of a geometric series is denoted as \(a\), and the common ratio of the series is \(r\). The sum of the first \(n\) terms \(S_n\) can be calculated using the formula:

• \( S_n = a \frac{1 - r^n}{1 - r} \)

This formula gives us a concise way to find the sum, even if \(n\) (the number of terms) is quite large. Remember, this formula works efficiently as long as \(req1\). Understanding this formula allows you to grasp more complex series concepts quickly!

Common Ratio

The common ratio \(r\) is a critical part of any geometric series. It's a constant factor between consecutive terms in the series. Suppose the first term is \(a\), then the second term will be \(ar\), the third term \(ar^2\), and so on. So, the pattern is:

• First term: \(a\)
• Second term: \(ar\)
• Third term: \(ar^2\)
• nth term: \(ar^{n-1}\)

Knowing the common ratio helps in quickly determining any term in the series and in calculating the sum of terms using the sum formula. If \(|r| < 1\), the series will converge as \(n\) tends toward infinity, meaning the terms get smaller and sum approaches a fixed number. If \(|r| > 1\), the series will diverge, leading to terms growing larger.

Series Sum Notation

Series sum notation is a compact way to express the addition of a sequence of numbers. In the context of geometric series, it allows us to succinctly denote the sum of some part of a series, or the whole.For a geometric series, the notation for the sum of the first \(n\) terms is \(S_n\), as shown in the equation:

• \( S_n = a \frac{1 - r^n}{1 - r} \)

Additionally, when considering the sum of these sums, known as \(U_n\), we use:

• \( U_n = \sum_{k=1}^{n} S_k \)

This notation allows us to convey complex sums in a clear and efficient manner, making it easier to perform operations and find relationships between different series properties. It's a helpful tool for finding patterns or deriving further series formulas.