Question

In the nth row of the triangle (a) last term \(=\) sum of the cubes of the first \(\mathrm{n}\) natural numbers (b) first term \(=\frac{\left(\mathrm{n}^{2}-\mathrm{n}+2\right)^{2}}{4}\) (c) positive square root of the first term is one more than the sum of first \(\mathrm{n}-1\) natural numbers (d) sum of the elements in the nth row \(=\frac{n\left(n^{2}+2\right)\left(3 n^{2}+1\right)}{12}\)

Short answer

Answer: The expression for the sum of the elements in the nth row of Triangle T based on property (d) is: \(\frac{n\left(n^2+2\right)\left(3n^2+1\right)}{12}\).

Step-by-step solution

Property (a): Last Term of the nth Row

According to this property, the last term in the nth row is equal to the sum of the cubes of the first n natural numbers. To analyze this, we'll use the formula for the sum of the first n cubes: Sum = \((1^3+2^3+3^3+\dots+n^3) = \left(\frac{n (n+1)}{2}\right)^2\) In the nth row, the last term of Triangle T will be equal to the sum of the cubes of the first n natural numbers.

Property (b): First Term of the nth Row

According to this property, the first term in the nth row of Triangle T can be expressed as: First Term = \(\frac{\left(n^2 - n + 2\right)^2}{4}\) In the nth row, the first term of Triangle T will follow this expression.

Property (c): Positive Square Root of the First Term

In this property, it is mentioned that the positive square root of the first term of the nth row is one more than the sum of the first n-1 natural numbers. Sum of first n natural numbers is given by the formula: Sum = \(\frac{n(n-1)}{2}\) To find the square root of the first term, we take the square root of the expression for the first term from property (b): \(\sqrt{\frac{\left(n^2 - n + 2\right)^2}{4}}\) According to the property, \(\sqrt{\frac{\left(n^2 - n + 2\right)^2}{4}} = 1 + \frac{n(n-1)}{2}\)

Property (d): Sum of the Elements in the nth Row

Lastly, this property states that the sum of the elements in the nth row of Triangle T can be expressed using the following formula: Sum = \(\frac{n\left(n^2+2\right)\left(3n^2+1\right)}{12}\) In the nth row, the sum of all terms in Triangle T will follow this expression.

Key concepts

Sum of Cubes

The concept of the sum of cubes refers to the sum of the cubes of the first n natural numbers. This is an extension of basic number series where we add numbers, but instead, we add their cubes. The formula for calculating the sum of the cubes is \( (1^3+2^3+3^3+...+n^3) = \left(\frac{n (n+1)}{2}\right)^2 \). It's interesting to note that the sum of cubes formula gives us a perfect square.This expression is useful in various mathematical problems and known for its simplicity and symmetry. - For example, if you take the first two natural numbers, 1 and 2, and calculate their cubes (1 and 8) and sum them, you will get 9, which is the square of 3.- Similarly, the three numbers 1, 2, and 3 give cubed values of 1, 8, and 27, and their sum is 36, which is the square of 6. Using this formula can make determining the sum of such numbers much simpler without manually cubing and adding each individually.

Series and Sequences

Understanding series and sequences is crucial when dealing with structured sets of numbers like those found in triangle number properties. A sequence is a list of numbers arranged in a particular order with a specific rule that dictates the progression of numbers. A series, on the other hand, is the sum of the terms of a sequence.In this context, we use the concepts of series and sequences to understand various properties of number triangles, such as the formula for the first term in the nth row.- The first term is determined by \( \frac{\left(n^2 - n + 2\right)^2}{4} \). This showcases how elements in the series relate to each other systematically.The systematic arrangements of numbers and manipulation of their elements in sequences and series form a foundation for algebra and offer paths to solving complex mathematical problems.In practical terms, these formulas allow mathematicians and students alike to predict number behaviors and recognize patterns that simplify calculations, especially when dealing with large sets of numbers.

Triangle Number Properties

Triangle numbers compose an interesting area of mathematics where numbers are represented in a triangular pattern. These numbers can be visualized as dots forming a triangular shape. The n-th triangle number is given by \( \frac{n(n+1)}{2} \), representing the sum of the first n natural numbers. This property is interesting because it connects linear sequence progressions with spatial representations, where each row adds a consecutive number of dots, building a new layer to the triangle.For instance, the third triangle number is 6, represented by three rows containing 1, 2, and 3 dots, respectively.- The number of dots, 6, corresponds to the sum \( 1+2+3 \).Triangle number properties extend into understanding more complex patterns and have applications in calculating combinatorial structures,as well as in solving problems involving geometric visuals and algebraic expressions.Together with series and sequences, they provide a framework for expressing and solving problems in mathematical induction and various levels of analysis. They reveal connections between algebraic expressions and geometric shapes, enriching both fields of study.