Problem 6 - A Complete Resource Book in Mathematics for JEE Main

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

The correct option is (C).

Step 1: Understanding the Series

The series is composed of terms like 1, 22, 333, ..., where each number is repeated 'n' times. For example, the 1st term is just 1, the 2nd term is 22, and the 9th is 999999999.

Step 2: Expression for nth Term

The nth term of this series can be represented as \(a_n = \frac{n \times (10^n - 1)}{9}\) because it's n occurrences of the digit 'n'. This can also be written as repeating the digit 'n', 'n' times, which sums up to \(n \times (10^{n-1} + 10^{n-2} + ... + 10^0)\).

Step 3: Sum of n Terms Expression

The sum of the first 'n' terms \(S_n\) can be expressed as \(S_n = \sum_{k=1}^{n} \frac{k(10^k - 1)}{9}\). This expression helps sum up each sequence step by step.

Step 4: Difference Expression

Compute \(S_n - S_{n-1}\). So \(S_n - S_{n-1} = a_n = \frac{n(10^n - 1)}{9}\). Simplifying gives \( \frac{n(10^n - 1)}{9}\) which matches \(9(S_n - S_{n-1}) = n(10^n - 1)\).

Step 5: Verify Given Options

Based on the difference expression \(9(S_n - S_{n-1}) = n(10^n - 1)\), verify with each given option. Option C precisely matches the derivation: \(9(S_n - S_{n-1}) = n(10^n - 1)\). The options A and B do not match the right-hand side entirely.

Key Concepts

Arithmetical Progression

Arithmetical progression is a sequence of numbers in which each term after the first is obtained by adding a fixed number to the previous term. This fixed number is called the 'common difference'. In an arithmetic sequence, you only need to know the initial term and the common difference to find any term. For example, in the series 2, 5, 8, 11, ..., the common difference is 3 and the first term is 2. This concept helps in predicting and calculating future terms in the sequence accurately.
In the given problem, although the series is not a typical arithmetic progression because each term increases by a concatenation of digits, understanding this concept helps in deducing possible connections between successive terms.

Sum of Terms

The sum of terms in a sequence or a series refers to the total obtained by adding the individual terms. In arithmetic progression, the sum of the first 'n' terms, denoted as \(S_n\), can be easily calculated through a formula: \[S_n = \frac{n}{2} \times (2a + (n-1)d)\]where \(a\) is the first term and \(d\) is the common difference.
However, the given problem differs slightly as it features a series formed by repeating digits. The sum in the given case is expressed with the formula:\[S_n = \sum_{k=1}^{n} \frac{k(10^k - 1)}{9}\] This formula was constructed considering the specific pattern of digits and their repetition in the series. Unlike simple series, this requires understanding how each term is formed and repeatedly added to derive the sum correctly.

Nth Term Calculation

The nth term of a series represents the specific element at position 'n'. Calculating this term is crucial for understanding and building upon the pattern in a series. For arithmetic progressions, it is done using the formula:\[a_n = a + (n-1)d\]where \(a\) is the first term and \(d\) is the common difference.In contrast, the series in this problem involves a unique nth term calculation:\[a_n = \frac{n \times (10^n - 1)}{9}\]This formula is derived because each term is the digit 'n', repeated 'n' times when broken down. Therefore, understanding and applying the correct formula are vital for accurate calculation of any specific term in specialized sequences like this one.

Problem Solving

Problem solving in mathematical series involves understanding patterns, recognizing formulas, and accurately applying them to find solutions. Here's how this is navigated:

• Comprehending the problem by clearly outlining given data and identifying what is being asked.
• Analyzing the sequence to determine if it fits known types of series, such as arithmetic or geometric.
• Applying the right formula for the sum of terms or nth term, specific to the series under study. This might mean deducing unique expressions for specialized sequences.
• Verifying your solution by cross-referencing with possible answer choices or through simplification to best ensure accuracy.

For the problem you're tackling, after understanding the sequence's unique pattern, the task was to compute expressions for consecutive terms and their sums. Following a step-by-step approach guarantees a thorough grasp and precise solution.