Question

The cubes of the natural numbers are grouped as \(1^{3},\left(2^{3}, 3^{3}\right),\left(4^{3}, 5^{3}, 6^{3}\right), \ldots\) then the sum of the numbers in the \(n\)th group is : (a) \(\frac{n^{3}}{12}\left(n^{2}+2\right)\left(n^{2}+4\right)\) (b) \(\frac{1}{8} n^{3}\left(n^{2}+1\right)\left(n^{2}+3\right)\) (c) \(\frac{n^{3}}{6}\left(n^{2}+1\right)\left(n^{2}+2\right)\) (d) none of these

Short answer

Answer: The sum of the numbers in the \(n\)th group is \(\frac{n^3}{6}\left(n^{2}+1\right)\left(n^{2}+2\right)\).

Step-by-step solution

Identify the pattern in the groupings

In the given sequence, we can see that the first group has 1 term, the second group has 2 terms, the third group has 3 terms, and so on. This pattern indicates that the \(n\)th group will have \(n\) terms.

Find the range of terms in the \(n\)th group

We can determine the range of terms present in the \(n\)th group by adding up the group sizes of the previous groups. For instance, the size of the first group is 1, the size of the second group is 2, and so on. Until the \(n\)th group, there are \(\frac{n(n-1)}{2}\) terms in total. Thus, the range of terms in the \(n\)th group would be from \(\frac{n(n-1)}{2}+1\) to \(\frac{n(n+1)}{2}\).

Calculate the sum of the cubes in the \(n\)th group

To find the sum of the cubes in the \(n\)th group, we can use the following formula for the sum of cubes of first \(k\) natural numbers: \(\sum_{i=1}^{k} i^3=\left[\frac{k(k+1)}{2}\right]^2\). We apply this formula to find the sum of cubes until the last term in the \(n\)th group, which is \(k = \frac{n(n + 1)}{2}\), and then subtract the sum of cubes until the term before the first term in the \(n\)th group, which is \(k = \frac{n(n - 1)}{2}\). Thus, the sum of cubes in the \(n\)th group is: $$\left(\frac{n(n+1)}{2}\right)^4 - \left(\frac{n(n-1)}{2}\right)^4$$

Simplify the expression and compare with the given options

We can factor out \(n^3\) from this expression to simplify it: $$n^3\left[\left(\frac{n+1}{2}\right)^4 - \left(\frac{n-1}{2}\right)^4\right]$$ Now we need to expand and simplify the terms inside the square brackets: $$n^3\left(\frac{n^4 + 4n^3 + 6n^2 + 4n + 1}{16} - \frac{n^4 - 4n^3 + 6n^2 - 4n + 1}{16}\right)$$ $$n^3\left(\frac{8n^3 + 8n}{16}\right)$$ $$\frac{n^3}{2}(n^2 + 1)(n^2 + 2)$$ Comparing this expression with the given options, we can see that it matches with option (c). Therefore, the sum of the numbers in the \(n\)th group is given by: $$\frac{n^3}{6}\left(n^{2}+1\right)\left(n^{2}+2\right)$$

Key concepts

Arithmetic Sequences

Arithmetic sequences are one of the simplest forms of mathematical patterns, characterized by a constant difference between consecutive terms. This difference, known as the common difference, remains the same throughout the sequence. For example, in the sequence 2, 4, 6, 8, ..., the common difference is 2.

Understanding arithmetic sequences can significantly help in recognizing patterns when grappling with exercises involving calculations of grouped numbers, like in our cube groupings exercise. In such cases, even though the numbers squared are not in arithmetic sequence, the number of terms in each group is - we have 1 term in the first group, 2 terms in the second, and so on.

Sum of Natural Numbers

The sum of natural numbers is a foundational concept in mathematics, and it plays a vital role in a variety of applications, from basic algebra to complex calculus. The natural numbers are the set of positive integers starting from 1. The formula for the sum of the first 'n' natural numbers is given by \( \frac{n(n+1)}{2} \). This formula is useful for finding the range of terms in sequences and series, as performed in Step 2 of our cube sum exercise.

For instance, knowing this can help understand that the sum of the first 'n' terms can be used to calculate the boundary of the 'n'th group in the sequence of cubes. This insight is instrumental for finding the sum of cubes in the group, enabling a student to approach problems breaking them down into more manageable parts.

Geometric Patterns

Geometric patterns arise from geometric figures and can be identified when specific mathematical structures follow a set of rules or formulas. Unlike arithmetic sequences that increase by the same amount, geometric patterns often involve multiplication or division by a constant. This kind of relationship between terms of a sequence is fundamental to understanding geometric sequences and series.

In the context of cubes in sequences, while the exercise doesn't involve a classic geometric series, recognising geometric patterns in mathematical problems can guide students in finding connections between numbers and shapes, subsequently leading to recognizing underlying mathematical principles at play. For example, visualizing the cube of a number as a geometric cube can aid in understanding the concept of raising a number to the third power.

Mathematical Series

Mathematical series involve the summation of the terms of a sequence. When these terms follow a specific rule, they form a series, and the sum can often be calculated using a formula. The term 'series' is particularly relevant to our cube sum exercise in Step 3, where the sum of cubes of the natural numbers is expressed as a series and calculated using the known sum formula \( \sum_{i=1}^{k} i^3 = \left[ \frac{k(k+1)}{2} \right]^2 \).

This special formula for the sum of cubes links to the broader concept of mathematical series - where an understanding of how to aggregate a set of numbers in a defined pattern can be crucial. It aids in finding succinct expressions for what might otherwise be a long and tedious sum, demonstrating the powerful shortcuts that series offer in the realm of mathematics.