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Chapter 4: Problem 5

Find an explicit formula for \(a_{n}\) where \(a_{1}=1\) and \(a_{n}=a_{n-1}+n\) for \(n \geq 2\).

Short Answer

Expert verified
The explicit formula for \(a_n\) is \(\frac{n(n+1)}{2}\).

Step by step solution

01

Understand the Recurrence Relation

The given sequence is a recurrence relation. We know that the first term is \(a_1 = 1\) and the rule \(a_n = a_{n-1} + n\) applies for \(n \geq 2\). Our goal is to find an explicit formula for \(a_n\), which means we need an expression that gives \(a_n\) directly without referring to previous terms.
02

Calculate Initial Terms

Let's calculate the first few terms to identify a pattern. We have: \(a_1 = 1\), \(a_2 = a_1 + 2 = 3\), \(a_3 = a_2 + 3 = 6\), and \(a_4 = a_3 + 4 = 10\). Observing these terms, we see the sequence: 1, 3, 6, 10,... which are triangular numbers.
03

Recognize the Pattern as Triangular Numbers

The terms \(1, 3, 6, 10,\ldots\) correspond to triangular numbers. Triangular numbers have the formula \(T_n = \frac{n(n+1)}{2}\), where \(T_n\) represents the nth triangular number. Let's test this formula with the terms we calculated: \(T_1 = 1\), \(T_2 = 3\), \(T_3 = 6\), and \(T_4 = 10\). These match our sequence terms.
04

Write the Explicit Formula

Since the sequence matches the triangular numbers, we identify that \(a_n\) follows the triangular numbers formula. Thus, the explicit formula for \(a_n\) is \(a_n = \frac{n(n+1)}{2}\). This formula gives the nth term directly in terms of n, without referring to previous terms.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Triangular Numbers
Triangular numbers are a fascinating and easily recognizable sequence of numbers. They are called triangular because they can be represented visually as a triangle. Imagine arranging dots to form a triangle. The number of dots in each row increases by one as you go down.
Each triangular number corresponds to a partial sum of the sequence of natural numbers: 1, 3, 6, 10, etc. For example, the first triangular number is 1, the second is 3 (1+2), the third is 6 (1+2+3), and so on. Each triangular number symbolizes a growing triangle of points where each new row adds an extra dot.
These numbers are described mathematically with the formula \( T_n = \frac{n(n+1)}{2} \), where \( T_n \) is the nth triangular number, showing how they are sums of sequential integers.
Explicit Formula
An explicit formula is a mathematical expression that allows you to calculate any term in a sequence directly. In the context of our original problem, finding this formula means you're able to identify the nth term without first calculating all the previous ones.
For the sequence given by the recurrence relation \( a_n = a_{n-1} + n \) with \( a_1 = 1 \), recognizing the pattern of triangular numbers helps to derive the explicit formula.
The explicit formula for this sequence is \( a_n = \frac{n(n+1)}{2} \). With this formula, you just plug in the value of \( n \) to obtain \( a_n \). It's like having a shortcut to directly reach the nth term, speeding up your calculations significantly.
Sequence Patterns
Recognizing patterns is a vital skill in mathematics, especially in sequences. Patterns provide a path to understanding complex sequences by identifying regularities or repeating units. This can lead to the discovery of specific properties of the sequence, or in our case, the ability to create an explicit formula for it.
In our original exercise, we noted a pattern when we calculated the first few terms: 1, 3, 6, 10. Do you see how they increase? This increase displays a pattern of triangular numbers which starts not randomly, but very deliberately, increasing by successive integers.
Detecting this familiar sequence pattern allows us to predict the rest of the sequence without additional calculations. It's a guiding light that allows mathematicians and learners alike to unlock the behavior of sequences efficiently.

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Most popular questions from this chapter

For the following exercises, indicate whether each of the following statements is true or false. If the statement is false, provide an example in which it is false.If \(b_{n} \geq 0\) is decreasing, then \(\sum_{n=1}^{\infty}\left(b_{2 n-1}-b_{2 n}\right)\) converges absolutely.

The following alternating series converge to given multiples of \(\pi .\) Find the value of \(N\) predicted by the remainder estimate such that the Nth partial sum of the series accurately approximates the left-hand side to within the given error. Find the minimum \(N\) for which the error bound holds, and give the desired approximate value in each case. Up to 15 decimals places, \(\pi=3.141592653589793 .\)[T] The Euler transform rewrites \(S=\sum_{n=0}^{\infty}(-1)^{n} b_{n}\) as \(S=\sum_{n=0}^{\infty}(-1)^{n} 2^{-n-1} \sum_{m=0}^{n}\left(\begin{array}{c}n \\ m\end{array}\right) b_{n-m}\). For the alternating harmonic series, it takes the form \(\ln (2)=\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n}=\sum_{n=1}^{\infty} \frac{1}{n 2^{n}} .\) Compute partial sums of \(\sum_{n=1}^{\infty} \frac{1}{n 2^{n}}\) until they approximate \(\ln (2)\) accurate to within \(0.0001\). How many terms are needed? Compare this answer to the number of terms of the alternating harmonic series are needed to estimate \(\ln (2)\).

In the following exercises, use an appropriate test to determine whether the series converges. $$ a_{k}=1 /\left(\begin{array}{l} 2 k \\ k \end{array}\right) $$

In the following exercises, use an appropriate test to determine whether the series converges. $$ \sum_{n=1}^{\infty} \frac{(n+1)}{n^{3}+n^{2}+n+1} $$

Use the root test to determine whether \(\sum_{m=1}^{\infty} a_{n}\) converges, where \(a_{n}\) is as follows. $$ a_{n}=\frac{(\ln (1+\ln n))^{n}}{(\ln n)^{n}} $$
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