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Chapter 13: Problem 59

Express each sum using summation notation. \(1+2+3+\cdots+20\)

Short Answer

Expert verified
\(\sum_{n=1}^{20} n\)

Step by step solution

01

- Identify the Sequence

Observe the given sequence: 1, 2, 3, ..., 20. It is an arithmetic sequence where each term increases by 1.
02

- Define the General Term

The general term of this arithmetic sequence can be expressed as: \(a_n = n\), where \(n\) represents the position of the term within the sequence.
03

- Determine the Range of the Summation

The sequence starts at 1 and ends at 20. Therefore, the index \(n\) ranges from 1 to 20.
04

- Construct the Summation Expression

Combine the general term and the range of the summation to express the sum in summation notation: \(\sum_{n=1}^{20} n\).

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Key Concepts

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

Arithmetic Sequence
An arithmetic sequence is a sequence of numbers in which the difference between successive terms is constant. This difference is called the common difference. For example, in the sequence 1, 2, 3, ..., 20, each term increases by 1. So, the common difference here is 1.

Here are some key features of an arithmetic sequence:
  • First term \(a_1\)
  • Common difference \(d\)
Understanding these features helps us describe the terms of the sequence clearly.
General Term
The general term of an arithmetic sequence gives us a formula to find any term based on its position (or index) in the sequence. For any arithmetic sequence, the general term is given by: \[ a_n = a_1 + (n - 1) \cdot d \] Where:
  • \(a_n\) is the \(n\)th term
  • \(a_1\) is the first term
  • \(d\) is the common difference
  • \(n\) is the term's position in the sequence
For the sequence 1, 2, 3, ..., 20, the first term \(a_1 = 1\), and the common difference \(d = 1\). Plugging in the values, we get: \[ a_n = 1 + (n - 1) \cdot 1 = n \] Therefore, the general term for this sequence is \(a_n = n\).
Index Range
The index range specifies the starting and ending values for the indices of the terms you're summing. For example, in the sequence 1 to 20, the index starts at 1 and ends at 20.

In summation notation, if \(n\) is the index variable:
  • The starting value of the index is the lower limit
  • The ending value of the index is the upper limit
For our example sequence, the index \(n\) ranges from 1 to 20. Therefore, the index range is from 1 to 20.
Summation Expression
A summation expression compactly represents the sum of a sequence. It鈥檚 written using the Greek letter sigma \((\sum)\), which denotes summation. The general format is: \[ \sum_{i=m}^{n} f(i) \] Where:
  • \( i \) is the index of summation
  • \( m \) is the lower limit
  • \( n \) is the upper limit
  • \( f(i) \) is the function of \( i \) giving the terms of the sequence
For our sequence 1, 2, 3, ..., 20, using \( n \) as the index, the summation expression is: \[\sum_{n=1}^{20} n \]

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Most popular questions from this chapter

Use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers \(n\). $$ \frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\frac{1}{3 \cdot 4}+\cdots+\frac{1}{n(n+1)} \equiv \frac{n}{n+1} $$

Show that the statement \(" n^{2}-n+41\) is a prime number" is true for \(n=1\) but is not true for \(n=41\).

Use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers \(n\). $$ n^{3}+2 n \text { is divisible by } 3 $$

A pendulum swings through an arc of length 2 feet. On each successive swing, the length of the arc is 0.9 of the previous length. (a) What is the length of the arc of the 10 th swing? (b) On which swing is the length of the arc first less than 1 foot? (c) After 15 swings, what total length has the pendulum swung? (d) When it stops, what total length has the pendulum swung?

Use mathematical induction to prove that $$ \begin{aligned} a+(a+d)+(a+2 d) & \\ +\cdots+[a+(n-1) d] &=n a+d \frac{n(n-1)}{2} \end{aligned} $$
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