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

Question

Statement-1: The sum of the series \(1+(1+2+4)+\) \((4+6+9)+(9+12+16)+\ldots+(361+380+400)\) is \(8000 .\) \(\begin{array}{l}\text { Statement-2: } \\ \text { number } n .\end{array}_{k=1}^{n}\left(k^{3}-(k-1)^{3}\right)=n^{3}\), 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

(B) Statement-1 is true, statement-2 is true; statement-2 is a correct explanation for Statement-1.

Step 1: Understanding the Series

The given series is split into several groups: \(1\), \((1+2+4)\), \((4+6+9)\), \((9+12+16)\), and continues in a similar pattern until \((361+380+400)\). Each group appears to follow a pattern based on perfect squares or cubes, and we need to identify it to find the sum.

Step 2: Recognizing Patterns

Observe the sequence of numbers in each group: \(1\), then \(1+2+4\) which is based on \(1^2, 2^2, 3^2\), and so on. The next group \(4+6+9\) corresponds to \(n^2 - (n-1)^2\) for increasing \(n\). Identify this pattern is based on differences of consecutive cubes.

Step 3: Using Statement-2

Statement-2 states that \(\sum_{k=1}^{n} \left(k^3 - (k-1)^3\right) = n^3\). This matches the pattern found. Each term represents \((k^2 + k(k-1) + (k-1)^2)\), which accumulates to \(k^3 - (k-1)^3\). This confirms the series follows this sum form.

Step 4: Calculating Sum

Calculate the entire series using Statement-2: for the last term \((361+380+400)\), we deduce this corresponds to the calculation for \(20^2\) as the last term \(n = 20\). By Statement-2, since \(400 = 20^2\), the sum is calculated as \(20^3 = 8000\). This agrees with the given series sum.

Step 5: Conclusion on Statements

Statement-1 is true as verified through calculation. Statement-2 also holds true and provides the correct explanation of how the series is constructed and summed, therefore verifying the series sums to 8000 as precisely described.

Key Concepts

Series Sum

A series sum refers to the result you get when you add up all the terms in a series. In the case of the given series, it is essential to identify and understand the individual patterns that make up the successive terms. By grasping the grouping of numbers such as \(1\), \((1+2+4)\), and so forth, you can see how these numbers fit into larger frameworks.

• Each group in this series stems from mathematical patterns (such as perfect squares or cubes) and leads to a systematic way of adding numbers.
• To find the series sum, one must correctly identify these intrinsic patterns and then apply formulas, like those evidenced in Statement-2, which provide shortcuts to summing the entire series.
• The calculated sum of the series in the original exercise is 8000, achieved by applying the correct formulas over patterns recognized in the numbers.

Understanding how each segment of the series works helps frame the series sum in a more approachable and logical manner.

Cubic Patterns

Cubic patterns in mathematics relate to sequences where numbers can be expressed through the cubes of integers. Recognizing these patterns can simplify the calculation of sums in series. In this exercise, each group of numbers derives from cube-related formulas.

• The underlying pattern suggests that each group's sum is derived from cube differences, specifically \(k^3 - (k-1)^3\).
• This means that when we look at sequences like \(1 + 2 + 4\) and analyze them through the concept of cubes, we see these aren't random numbers but structured outcomes of cubic calculations.
• Cubic patterns help form a concrete understanding of the group's formula based on increments of cubes, a fascinating aspect that simplifies summing series.

Utilizing these patterns allows for a deeper understanding of how sequences build up and make the calculation of large sums manageable.

Mathematical Proof

Mathematical proof provides the groundwork for verifying statements in mathematics by using logical sequences and established formulas. In solving our exercise, Statement-2 functions as a mathematical proof that shows the truth of the summation technique.

• It claims that for any natural number \(n\), the series of differences of consecutive cubes \( (k^3 - (k-1)^3) \) sums to \( n^3 \).
• This proof affirms how each calculated term accumulates precisely to the right sum, demonstrating not only the truth of the statements in the exercise but how these sums apply mathematically.
• Such proofs provide robust tools in mathematics to validate the structure and results of complex series.

Having a solid mathematical proof reassures students of the consistency and correctness of their calculations, enriching their comprehension of series and summation.