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Chapter 10: Problem 45

(a) write using summation notation, and (b) find the sum. $$a+2 a+3 a+\cdots+60 a$$

Short Answer

Expert verified
The sum of the series is given by \(1830a\). The series in summation notation can be expressed as \(a \cdot \sum_{i=1}^{60} i\).

Step by step solution

01

Write Using Summation Notation

To write this sequence in summation notation we use the Greek letter Sigma (\(\Sigma\)) that's commonly used to denote summation. The term \(a\), being constant for each term of the sequence, can be written outside the summation. The variable of summation is typically represented as \(i\) and in this case varies from 1 to 60. So, the sequence can be written as: \[ a \cdot \sum_{i=1}^{60} i \]
02

Find the Sum

In order to find the sum of the sequence, we use the formula for the sum of the first \(n\) natural numbers, which is given by \(\frac{n \cdot (n+1)}{2}\). Here, \(n = 60\). So, the sum of the sequence is: \[ a \cdot \frac{60 \cdot (60+1)}{2} = 1830a \]

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Key Concepts

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

Understanding Sequences
A sequence is a set of numbers arranged in a specific order. In mathematics, sequences are ordered lists of numbers that follow a particular pattern. When you look at the series given in the exercise, it represents an arithmetic sequence. An arithmetic sequence is one where the difference between consecutive terms is constant. This constant difference is known as the common difference.

The given sequence, \( a, 2a, 3a, \ldots, 60a \), shows that each term increases by the same amount, which is \( a \) in this case. Here, the first term \( (a_1) \) is \( a \), and the common difference \( (d) \) is also \( a \).
  • This pattern helps in writing the sequence succinctly using summation notation.
  • An important characteristic of sequences is their limit; in this case, the sequence ends at \( 60a \).
When sequences are summed up, they form a series, which is what we've been tasked with finding: the sum of this particular series.
Exploring Natural Numbers
Natural numbers are a set of positive integers starting from 1, extending infinitely. This set includes numbers like 1, 2, 3, and so on. In the context of our exercise, these natural numbers play a vital role in defining the series that is represented.

The sequence described in the exercise uses natural numbers to define the multiple of \( a \) for each term. For each term \( i \) in our series, there is a corresponding value in the natural number set, starting from 1 and ending at 60.
  • Natural numbers are the backbone for constructing the majority of arithmetic sequences.
  • They provide a simple, clear pattern for constructing sequences, where each term follows the previous by a fixed addition.
Understanding the application of natural numbers assists in comprehending why the given formula for the sum of natural numbers helps find the sum of such sequences.
Delving into Sigma Notation
Sigma notation, denoted by the Greek letter \( \Sigma \), is used to represent the summation of sequences concisely. It is a powerful tool that simplifies the expression of long sums into a single, manageable mathematical statement.

In the exercise, we utilized this notation to sum the sequence \( a , 2a, 3a, \ldots, 60a \). Here's how it works:
  • \( \Sigma \) denotes the sum of a sequence.
  • Below the sigma, the initial point or lower bound of the sequence is placed, such as \( i=1 \).
  • Above the sigma, the endpoint or upper bound is noted, such as 60, indicating the sequence ends there.
  • The expression to the right of the sigma describes each term, often in terms of a variable, in this case, \( i \).
For our series, the notation \( a \cdot \sum_{i=1}^{60} i \) indicates multiplying each natural number from 1 to 60 by \( a \) and summing those products. This compact notation transforms the representation of the series from long form to a brief, comprehensible math formula.

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Most popular questions from this chapter

In Exercises \(5-25,\) prove the statement by induction. $$\begin{aligned} &1 \cdot 2+2 \cdot 3+3 \cdot 4+\dots+n(n+1)\\\ &=\frac{n(n+1)(n+2)}{3} \end{aligned}$$

Recreation The following table gives the amount of money, in billions of dollars, spent on recreation in the United States from 1999 to \(2002 .\) (Source: Bureau of Economic Analysis) $$\begin{aligned} &\text { Year } \quad 1999 \quad 2000 \quad 2001 \quad 2002\\\ &\begin{array}{l} \text { Amount } \\ \text { (S billions) } 546.1 \quad 585.7 \quad 603.4 \quad 633.9 \end{array} \end{aligned}$$ Assume that this sequence of expenditures approximates an arithmetic sequence. (a) If \(n\) represents the number of years since 1999 , use the linear regression capabilities of your graphing calculator to find a function of the form \(f(n)=a_{0}+n d, n=0,1,2,3, \ldots,\) that models these expenditures. (b) Use your model to project the amount spent on recreation in 2007

Answer True or False. Consider randomly picking a card from a standard deck of 52 cards. The complement of the event "picking a black card" is "picking a heart."

In this set of exercises, you will use sequences to study real-world problems. A sequence of square boards is made as follows. The first board has dimensions 1 inch by 1 inch, the second has dimensions 2 inches by 2 inches, the third has dimensions 3 inches by 3 inches, and so on. (a) What type of sequence is formed by the perimeters of the boards? Explain. (b) Write a rule for the sequence formed by the areas of the boards. Is the sequence arithmetic, geometric, or neither? Explain your answer.

In the board game Mastermind, one of two players chooses at most four pegs to place in a row of four slots, and then hides the colors and positions of the pegs from his opponent. Each peg comes in one of six colors, and the player can use a color more than once. Also, one or more of the slots can be left unfilled. (a) How many different ways are there to arrange the pegs in the four-slot row? In this game, the order in which the pegs are arranged matters. (b) The Mastermind website states: "With 2401 combinations possible, it's a mind-bending challenge every time!" Is combination the appropriate mathematical term to use here? Explain. This is an instance of how everyday language and mathematical language can be contradictory. (Source: www.pressman.com)
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