Problem 60 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Let \(S_{n}(1 \leq n \leq 9)\) denotes the sum of \(n\) terms of series \(1+22+333+\ldots+999999999\), then for \(2 \leq n \leq 9\) (A) \(S_{n}-S_{n-1}=\frac{1}{9}\left(10^{n}-n^{2}+n\right)\) (B) \(S_{n}=\frac{1}{9}\left(10^{n}-n^{2}+2 n-2\right)\) (C) \(9\left(S_{n}-S_{n-1}\right)=n\left(10^{n}-1\right)\) (D) None of these

Step-by-Step Solution

Answer

Option C is correct: \(9(S_n - S_{n-1}) = n(10^n - 1)\).

Step 1: Understanding the Series

The series is described as \(1 + 22 + 333 + \ldots + \text{(n repeated n times)}\). For a general term in position \(n\), it is represented as repetition of digit \(n\) for \(n\) times, which can be expressed as \(\frac{n}{9} \times (10^n - 1)\).

Step 2: Recognizing Patterns for Sums

For \(S_n\), the sum of the first \(n\) terms of the given series is \(S_n = 1 + (22) + (333) + \ldots + \left( \frac{n}{9} \times (10^n - 1)\right)\). The goal is to determine the difference \(S_n - S_{n-1}\) to validate or invalidate options.

Step 3: Calculating Difference \(S_n - S_{n-1}\)

Given the general form for each term, compute \(S_n - S_{n-1}\):\[S_n - S_{n-1} = \frac{n}{9} \times (10^n - 1).\] This is computed by subtracting the sum of the first \((n-1)\) terms from the sum of the first \(n\) terms.

Step 4: Validate Each Option

Let's compare:- \(\text{Option A}\) expects \(\frac{1}{9}(10^n - n^2 + n)\)- \(\text{Option B}\) gives an expression for \(S_n\) itself, not the difference.- \(\text{Option C}\) states \(9(S_n - S_{n-1}) = n(10^n - 1)\)Substituting \(S_n - S_{n-1}\) from Step 3 into the options:- For \(\text{Option A}\), it doesn't match \(\frac{n}{9} \times (10^n - 1)\).- For \(\text{Option C}\), multiply by 9: \[9 \times \left( \frac{n}{9} \times (10^n - 1) \right) = n(10^n - 1)\] which matches exactly with option C.

Key Concepts

Summation

In mathematics, summation refers to the process of adding together a sequence of numbers. This concept is essential when dealing with series, where you need to find the total of a sequence of terms. In the given problem, we are considering a series where each term consists of numbers repeated multiple times, such as 1, 22, 333, and so on. The summation for these terms is expressed as \(S_n\), representing the sum of the first \(n\) terms. To calculate \(S_n\), you sum up all individual terms from the start of the series up to the nth term.
Calculating summations is vital in understanding series because it helps to find the total value efficiently instead of adding each term one by one. This process often involves recognizing a pattern or a formula that can make finding the sum of the terms simpler. In mathematical series like our example, you might use explicit formulas, such as \(S_n = 1 + 22 + 333 + \ldots + (\frac{n}{9} \times (10^n - 1))\), to derive these sums. Understanding how to handle summations can simplify solving complex mathematical problems, especially those involving series.

Difference Calculation

The concept of difference calculation involves determining the difference between two consecutive terms or sums, which is a crucial operation in sequences and series. In this exercise, finding \(S_n - S_{n-1}\) helps us understand how each new term affects the total sum as you progress through the sequence.
When you subtract \(S_{n-1}\) from \(S_n\), you essentially isolate the nth term of the series. This difference is critical when you want to verify solutions or confirm patterns, as shown in the step-by-step solution. In this problem, the difference \(S_n - S_{n-1}\) is expressed as \(\frac{n}{9} \times (10^n - 1)\).
This calculation method is not only computational but also analytical, allowing us to validate given options like option C correctly. By looking at the difference formula, you get a clearer view of the structure and specific behavior of the series as you increase \(n\), enhancing your understanding of the pattern across terms.

JEE Main Mathematics

Understanding series and their differences is often a key aspect of exams like JEE Main Mathematics, which tests students on their knowledge of various mathematical concepts. This problem demonstrates typical questions you might find in such examinations, where identifying patterns, calculating summations, and ensuring the correctness of derived formulas are essential skills.
The JEE Main Mathematics exam frequently includes questions about arithmetic series to test your critical thinking and problem-solving abilities. It's not just about knowing formulas but also about understanding their derivation and application to solve problems effectively. These skills are developed through exercises like this one, which sharpens your ability to deal with sequences, series, and different calculation techniques.
Practicing these types of problems prepares students for complex mathematical reasoning, necessary for success in competitive exams. As seen in this exercise, recognizing correct patterns and validating formulas is a vital part of the rigorous preparation required for the JEE Main Mathematics test.