Problem 57 - Algebra 2

Question

Geometry The triangular numbers form a sequence. The diagram represents the first three triangular numbers: \(1,3,\) and \(6 .\) a. Find the fifth and sixth triangular numbers. b. Write a recursive formula for the \(n\) th triangular number. c. Is the explicit formula \(a_{n}=\frac{1}{2}\left(n^{2}+n\right)\) the correct formula for this sequence? How do you know?

Step-by-Step Solution

Answer

The fifth and sixth triangular numbers are 15 and 21 respectively. The recursive formula for the nth triangular number is \(T_n = T_{(n-1)} + n\). The given explicit formula \(a_{n}=\frac{1}{2}\left(n^{2}+n\right)\) is indeed correct for this sequence.

Step 1: Find the fifth and sixth triangular numbers

Triangular numbers are the count of dots that can form an equilateral triangle. The nth triangular number is the number of dots in the triangular arrangement with n dots on a side, which could also be represented as the sum of the n natural numbers from 1 to n. To find the fifth triangular number, the numbers from 1 to 5 are added: \(1+2+3+4+5= 15\). Similarly, to find the sixth triangular number, the numbers from 1 to 6 are added: \(1+2+3+4+5+6= 21\).

Step 2: Formulate a recursive formula for the nth triangular number

A recursive formula defines the sequence based on one or more previous terms. For this sequence, the nth triangular number can be obtained by adding n to the (n-1)th triangular number. So, the recursive formula can be written as \(T_n = T_{(n-1)} + n\), where \(T_n\) is the nth triangular number and \(T_{(n-1)}\) is the triangular number before it.

Step 3: Confirm the validity of the explicit formula

The explicit formula provides the nth term directly without referring to the previous terms. The given explicit formula \(a_{n}=\frac{1}{2}\left(n^{2}+n\right)\) is correct for this sequence. This can be confirmed by testing it for values of n. For example, for n=5, \(a_{n}=\frac{1}{2}\left(5^{2}+5\right) = 15\), which validates that it's the fifth triangular number.

Key Concepts

Recursive Formula

A recursive formula allows us to find each term in a sequence by building off the terms that come before it. In the context of triangular numbers, the recursive formula answers the question: "How can we find the next triangular number given the previous one?"

For triangular numbers, this recursive relationship is expressed as follows:

• The nth triangular number is equal to the (n-1)th triangular number plus n.
• This is written as: \( T_n = T_{n-1} + n \)
• Where \( T_n \) represents the nth triangular number.

For instance, once you have calculated the 4th triangular number, which is 10, you can use the formula to find the 5th. You'd add 5 to 10, arriving at 15. This systematic approach helps build the series efficiently and logically.

Explicit Formula

An explicit formula is particularly useful as it allows one to jump straight to any term in a sequence without needing the preceding terms. In the case of triangular numbers, the explicit formula provides a direct calculation.

The familiar explicit formula for triangular numbers is:

• \( a_n = \frac{1}{2}(n^2 + n) \)

This formula tells us that the nth triangular number can be computed using basic arithmetic and algebraic operations. For example, to find the 6th triangular number:

• Insert the value of n (6) into the formula.
• Calculate the result: \( \frac{1}{2}(6^2 + 6) = 21 \).

This straightforward approach confirms its effectiveness by showing that both the explicit and recursive methods agree on the sequence outcomes, such as the 5th and 6th triangular numbers.

Sequence of Numbers

A sequence of numbers is an ordered list where each number is known as a "term." The triangular numbers sequence is a specific kind of sequence that represents numbers which can form an equilateral triangle when arranged as dots.

Some characteristics of this sequence include:

• It starts with 1 and increases as more terms are added: \( 1, 3, 6, 10, 15, ... \)
• Each term represents the sum of all natural numbers up to that position.

This sequence is a beautiful example of a foundational mathematical concept with clear geometric representation. Each step of the sequence builds on the last, forming clear visual triangles and maintaining a straightforward pattern. Understanding these properties provides a deeper appreciation and comprehension of how the world of numbers can easily translate into a spatial form.