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Chapter 13: Problem 3

Fill in each blank with the correct response. If \(a_{n}=2 n,\) then \(a_{40}=\)________.

Short Answer

Expert verified
80

Step by step solution

01

Identify the General Formula

Recognize the given general formula for the sequence, which is defined as: \[a_{n} = 2n\]
02

Substitute the Given Value

Substitute the given term number, 40, into the formula. This means replacing \(n\) with 40: \[a_{40} = 2 \times 40\]
03

Simplify the Expression

Simplify the mathematical expression obtained by plugging in 40: \[a_{40} = 80\]

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Key Concepts

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

general formula
An arithmetic sequence is a sequence of numbers where each term increases by a constant amount. The general formula for an arithmetic sequence allows us to find any term in the sequence. For the given problem, the general formula is written as: explicit general formula The general formula for this specific exercise is: \[a_n = 2n\] This means that to find any term (\[a_n\]), you multiply the position of that term () by 2. The general formula is key because it creates a straightforward way to find any term in the sequence without needing to know the previous terms.
sequence term
The term of the sequence refers to any specific number within it. In our exercise, we are finding the 40th term of the sequence defined by the general formula \[a_n = 2n\]. Let's break this down:
  • The position of the term we are looking for is 40.
  • Substituting the position number (40) into the general formula: \[a_{40} = 2 \times 40\]
  • This calculation is the actual essence of finding the required term.
So, the sequence term at position 40 is 80.
substitution method
The substitution method is a key mathematical technique used to find specific values in formulas and equations. It involves:
  • Taking a known formula.
  • Replacing one variable with a specific given value.
For our exercise, we used the substitution method as follows:
  1. We identified the general formula: \[a_{n} = 2n\]
  2. We substituted the given value 40 for n in that formula: \[a_{40} = 2 \times 40\]
  3. We simplified the expression to find the answer: 80.
By substituting the term number 40 into our general formula, we efficiently found the term value.

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