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Chapter 4: Problem 3

Find the first six terms of each of the following sequences, starting with \(n=1\). $$ a_{1}=1 \text { and } a_{n}=a_{n-1}+n \text { for } n \geq 2 $$

Short Answer

Expert verified
The first six terms are 1, 3, 6, 10, 15, and 21.

Step by step solution

01

Understand the Sequence Definition

The sequence is defined recursively. We start with \(a_1 = 1\), and for \(n \geq 2\), each term \(a_n\) is calculated by adding \(n\) to the previous term \(a_{n-1}\).
02

Calculate the Second Term, \(a_2\)

Using the formula \(a_n = a_{n-1} + n\), substitute \(n = 2\), we get \(a_2 = a_1 + 2 = 1 + 2 = 3\).
03

Calculate the Third Term, \(a_3\)

Substitute \(n = 3\) in the formula: \(a_3 = a_2 + 3 = 3 + 3 = 6\).
04

Calculate the Fourth Term, \(a_4\)

Substitute \(n = 4\) in the formula: \(a_4 = a_3 + 4 = 6 + 4 = 10\).
05

Calculate the Fifth Term, \(a_5\)

Substitute \(n = 5\) in the formula: \(a_5 = a_4 + 5 = 10 + 5 = 15\).
06

Calculate the Sixth Term, \(a_6\)

Substitute \(n = 6\) in the formula: \(a_6 = a_5 + 6 = 15 + 6 = 21\).

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Key Concepts

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

Sequence Definition
In mathematics, a sequence is a list of numbers arranged in a specific order. Each number in this list is called a term. Sequences can be finite, meaning they have a definite number of terms, or they can be infinite, continuing indefinitely. A crucial aspect of sequences is how each term is related to others.

The term "recursive sequence" refers to sequences where each term is defined based on previous terms. This contrasts with explicit sequences which give a direct formula to find any term. Recursive sequences are like step-by-step instructions, analyzing each term is derived from its predecessor. Understanding these sequences helps us see the relationship between numbers, forming a foundation for many mathematical concepts.
Recursive Formula
The recursive formula is at the heart of recursive sequences. This formula helps dictate the rule for finding subsequent terms based on preceding ones. In the given example, we see:
  • The starting point, or base case, is where it all begins. Here, it starts with \(a_1 = 1\).
  • The recursive step explains how to get every following term. For \(n \geq 2\), \(a_n = a_{n-1} + n\) guides us.
This recursive definition ensures that with every increasing \(n\), you know how to find the next term. Recursive formulas empower mathematics to build complex and engaging patterns from simple beginnings. They challenge us to think one step at a time, making recursive logic a powerful tool for solving problems.
Term Calculation
Calculating terms in a recursive sequence involves replacing variables in the recursive formula with known values of previous terms. Let's walk through finding the first few terms:
  • Begin with \(a_1 = 1\) as given.
  • Use the recursive step for subsequent terms:
  • To find \(a_2\), use \(a_2 = a_1 + 2 = 1 + 2 = 3\).
  • For \(a_3\), calculate \(a_3 = a_2 + 3 = 3 + 3 = 6\).
  • Continuing this process for \(a_4, a_5,\) and \(a_6\) results in 10, 15, and 21 respectively.
This systematic approach highlights the beauty of recursive sequences. Each calculation builds on prior steps, offering both predictive insight and surprise as new terms emerge. Understanding term calculation evokes a sense of pattern appreciation in mathematics.
Mathematics Education
Recursive sequences are not just about numbers; they hold educational value in learning math. Exploring these sequences develops critical thinking and problem-solving skills. This hands-on activity allows students to apply logic in a stepwise fashion.

Understanding such sequences deepens comprehension of mathematical structures, preparing learners for more advanced topics like calculus and discrete mathematics. By analyzing how each term depends on the previous one, students foster an appreciation for detail and precision in their work. Learning recursive formulas paves the way for exploring more intricate mathematical phenomena, broadening the landscape of students' education.

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Most popular questions from this chapter

The following series do not satisfy the hypotheses of the alternating series test as stated. In each case, state which hypothesis is not satisfied. State whether the series converges absolutely.Show that the alternating series \(\frac{2}{3}-\frac{3}{5}+\frac{4}{7}-\frac{5}{9}+\cdots\) does not converge. What hypothesis of the alternating series test is not met?

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

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 alternating harmonic series converges because of cancellation among its terms. Its sum is known because the cancellation can be described explicitly. A random harmonic series is one of the form \(\sum_{n=1}^{\infty} \frac{S_{n}}{n}\), where \(s_{n}\) is a randomly generated sequence of \(\pm 1\) 's in which the values \(\pm 1\) are equally likely to occur. Use a random number generator to produce 1000 random \(\pm 1\) s and plot the partial sums \(S_{N}=\sum_{n=1}^{N} \frac{s_{n}}{n}\) of your random harmonic sequence for \(N=1\) to \(1000 .\) Compare to a plot of the first 1000 partial sums of the harmonic series.

A legend from India tells that a mathematician invented chess for a king. The king enjoyed the game so much he allowed the mathematician to demand any payment. The mathematician asked for one grain of rice for the first square on the chessboard, two grains of rice for the second square on the chessboard, and so on. Find an exact expression for the total payment (in grains of rice) requested by the mathematician. Assuming there are 30,000 grains of rice in 1 pound, and 2000 pounds in 1 ton, how many tons of rice did the mathematician attempt to receive?

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.Let \(a_{n}^{+}=a_{n}\) if \(a_{n} \geq 0\) and \(a_{n}^{-}=-a_{n}\) if \(a_{n}<0 .\) (Also, \(a_{n}^{+}=0\) if \(a_{n}<0\) and \(a_{n}^{-}=0\) if \(\left.a_{n} \geq 0 .\right)\) If \(\sum_{n=1}^{\infty} a_{n}\) converges conditionally but not absolutely, then neither \(\sum_{n=1}^{\infty} a_{n}^{+}\) nor \(\sum_{n=1}^{\infty} a_{n}^{-}\) converge.
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