Problem 42 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Statement-1: The sum of the series \(1+(1+2+4)+\) \((4+6+9)+(9+12+16)+\ldots+(361+380+400)\) is \(8000 .\) \(\begin{aligned}&\text { Statement-2: } \\\&\text { number } n .\end{aligned} \sum_{k=1}^{n}\left(k^{3}-(k-1)^{3}\right)=n^{3}\), for any natural (A) Statement- 1 is false, Statement- 2 is true. (B) Statement-1 is true, statement-2 is true; statement- 2 is a correct explanation for Statement- 1 (C) Statement-1 is true, statement-2 is true; statement-2 is not a correct explanation for Statement- 1 . (D) Statement- 1 is true, statement- 2 is false.

Step-by-Step Solution

Answer

(C) Statement-1 is true, Statement-2 is true, but not a correct explanation.

Step 1: Analyze Statement-1

The series given is \( 1 + (1+2+4) + (4+6+9) + (9+12+16) + \ldots + (361 + 380 + 400) \). It's broken down into sets of increasing cubes: \( 1^3 = 1 \), \((2^3 - 1^3) = 7\), \((3^3 - 2^3) = 19\), and so on.

Step 2: Evaluate Sum Structure in Statement-1

Look at the sequence of differences: \(1, 7, 19, \ldots\). These are form of \(k^3 - (k-1)^3\), where the nth term \(a_n = n^3 - (n-1)^3 = 3n^2 - 3n + 1\).

Step 3: Calculate the Total Number of Terms in Series

Since the series ends at \(400\) which is \(7^3\), the number of terms, \(n = 7\). Therefore, the sum of the series can be calculated.

Step 4: Sum Up Series to Verify Statement-1

Sum \( (1^3) + (2^3 - 1^3) + (3^3 - 2^3) + \ldots + (20^3 - 19^3) = 20^3 = 8000 \). Statement-1 appears true.

Step 5: Analyze Statement-2

Statement-2 is purely mathematical. We know \( \sum_{k=1}^{n} \left( k^3 - (k-1)^3 \right) = n^3 \) simplifies each term except the last. This is verified by computation and mathematical identities.

Step 6: Compare and Conclude

Statement-1 is computed to be true, and Statement-2 is mathematically true. However, Statement-2 does not explain Statement-1 since they are related but different concepts.

Key Concepts

Series Sum

Understanding the concept of *Series Sum* is crucial in evaluating the given problem. A series is essentially the sum of the terms of a sequence. The exercise presents a series featuring cubes of numbers. When analyzing series, particularly those involving cubed numbers, it’s important to recognize patterns or structures that can simplify calculations. In our series, each group of terms can be expressed as the difference between successive cubes:

• The term from one cube series minus the last term of the previous.
• This leads to an effective simplification of many terms.

Recognizing this pattern helps in computing the precise sum quickly, by just considering the initial and final cubes, which in this problem aligns to the last number cubed minus the sum of all lesser cubes. Breaking down a sum into such simplified components allows for easier computation and greater insight into the nature of the sequence itself.

Cubed Numbers

Cubed numbers, or numbers raised to the power of three, appear frequently in mathematics, particularly in the study of series and sequences. A cubed number can be written as \( n^3 \), where \( n \) is a positive integer. Understanding how these cubes interact and form differences is key to solving the exercise.In this problem, we observe the sequence of differences such as \((2^3 - 1^3)\), \((3^3 - 2^3)\), etc. These differences give rise to the series format:\( n^3 - (n-1)^3 \). Upon simplification, this results in the expression \( 3n^2 - 3n + 1 \), reflecting the basic algebraic manipulation of cubed numbers.Recognizing that many terms in such a series cancel each other out simplifies computations tremendously. This peculiar behavior of cubed numbers aids in evaluating entire series by focusing only on the edge cases of the cube differences. As you grow familiar with these transformations, your ability to spot such simplifications enhances dramatically.

Mathematical Identities

Mathematical identities serve as special equations that hold true for all values of their variables. They are exceptionally handy tools for simplifying complex expressions or verifying assumptions, like those in the exercise.The second statement in the problem, \( \sum_{k=1}^{n} \left( k^3 - (k-1)^3 \right) = n^3 \), is itself a classic identity. It's an elegant simplification that captures the nature of telescopic series, where most terms cancel out except those at the boundaries.

• This identity helps prove the sum sequence solution by alignment with known mathematical truths.
• It also showcases how cube-related identities can simplify large scale operations.

Using identities effectively demands a recognition of the recurring patterns in expressions. As you master them, you gain the ability to not only solve problems faster but also verify solutions quickly by checking their consistency with such unchanging truths.