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Chapter 15: Problem 13

In Exercises \(12-15\), write the first 5 terms of the sequence \(a_{n}=f(n)\) $$ f(t)=3 t-1 $$

Short Answer

Expert verified
Answer: The first 5 terms of the sequence are 2, 5, 8, 11, and 14.

Step by step solution

01

Function of the sequence given

For this exercise, we are given the function \(f(t) = 3t - 1\) which determines the terms of the sequence.
02

Finding the first term (n=1)

When substituting \(n=1\) into the function, we get: $$ a_1 = f(1) = 3(1) - 1 $$ Calculating the result, we find the first term: $$ a_1 = 2 $$
03

Finding the second term (n=2)

Now, substituting \(n=2\) into the function, we have: $$ a_2 = f(2) = 3(2) - 1 $$ Calculating the result, we find the second term: $$ a_2 = 5 $$
04

Finding the third term (n=3)

We continue substituting \(n=3\) into the function: $$ a_3 = f(3) = 3(3) - 1 $$ Calculating the result, we find the third term: $$ a_3 = 8 $$
05

Finding the fourth term (n=4)

Next, we substitute \(n=4\) into the function: $$ a_4 = f(4) = 3(4) - 1 $$ Calculating the result, we find the fourth term: $$ a_4 = 11 $$
06

Finding the fifth term (n=5)

Finally, we substitute \(n=5\) into the function: $$ a_5 = f(5) = 3(5) - 1 $$ Calculating the result, we find the fifth term: $$ a_5 = 14 $$
07

Writing down the first 5 terms

Based on our calculations, the first 5 terms of the sequence are: $$ a_1 = 2, a_2 = 5, a_3 = 8, a_4 = 11, a_5 = 14 $$

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Key Concepts

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

Linear Function
Linear functions are mathematical expressions that create a straight line when graphed. In simple terms, they have the form \( f(t) = mt + c \), where \( m \) represents the slope and \( c \) is the y-intercept. In our exercise, the function is \( f(t) = 3t - 1 \). This tells us several things:
  • The slope \( m \) is 3, meaning the sequence increases by 3 with each step.
  • The y-intercept \( c \) is -1, which is the starting point adjustment of the sequence on the y-axis.
Linear functions are incredibly useful in arithmetic sequences because they show how each term relates to the next. If you can understand the "slope" of these functions, you'll see how quickly things change as you move along the sequence.
Sequence Terms Calculation
Calculating the terms of a sequence involves systematically substituting values into the function for each term's position, known as \( n \). In the current problem, we are asked to find the first five terms for \( f(t) = 3t - 1 \). Here's how it works:
  • First Term: For \( n=1 \), substitute 1 into the function: \( a_1 = 3(1) - 1 = 2 \).
  • Second Term: For \( n=2 \), substitute 2: \( a_2 = 3(2) - 1 = 5 \).
  • Third Term: Continue with \( n=3 \): \( a_3 = 3(3) - 1 = 8 \).
  • Fourth Term: Substitute for \( n=4 \): \( a_4 = 3(4) - 1 = 11 \).
  • Fifth Term: Finally, for \( n=5 \): \( a_5 = 3(5) - 1 = 14 \).
This process demonstrates how each term is derived individually by plugging \( n \) into the linear function.
Mathematical Substitution
Astutely substituting values into equations is a foundational concept in mathematics. It involves placing a set value into a function to solve for another variable. In the sequence calculation, the function \( f(t) = 3t - 1 \) illustrates this elegantly:
  • Each term in the sequence \( a_n \) is represented by substituting \( n \) for \( t \) in the function.
  • This straightforward process requires settling input values one step at a time to yield outputs.
  • Our careful substitution turned \( f(t) \) into values like \( f(1) \), \( f(2) \), and so on, thus calculating the sequence terms effectively.
Mathematical substitution simplifies complex problems into manageable tasks, ensuring that each step correctly progresses towards the final answer.

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Most popular questions from this chapter

Evaluate the sums in Problems 5-12 using the formula for the sum of an arithmetic series. $$ -4.01-4.02-4.03-\cdots-4.35 $$

In Exercises 2-4, complete the tables with the terms of the arithmetic series \(a_{1}, a_{2}, \ldots, a_{n},\) and the sequence of partial sums, \(S_{1}, S_{2}, \ldots, S_{n} .\) State the values of \(a_{1}\) and \(d\) where \(a_{n}=a_{1}+(n-1) d\) The table for the terms of the series \(2,6,10, \ldots,\) is $$ \begin{array}{l|l|l|l|l|l|l|l|l} \hline n & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ \hline a_{n} & 2 & 6 & 10 & & & & & \\ \hline S_{n} & 2 & 8 & & & & & & \\ \hline \end{array} $$

A ball is dropped from a height of 20 feet and bounces. Each bounce is \(3 / 4\) of the height of the bounce before. Thus after the ball hits the floor for the first time, the ball rises to a height of \(20(3 / 4)=15\) feet, and after it hits the floor for the second time, it rises to a height of \(15(3 / 4)=20(3 / 4)^{2}=11.25\) feet (a) Find an expression for the height to which the ball rises after it hits the floor for the \(n^{\text {th }}\) time. (b) Find an expression for the total vertical distance the ball has traveled when it hits the floor for the first, second, third, and fourth times. (c) Find an expression for the total vertical distance the ball has traveled when it hits the floor for the \(n^{\text {th }}\) time. Express your answer in closed form.

Table 15.2 shows data of AIDS deaths \(^{6}\) that occurred in the US where \(a_{n}\) is the number of deaths in year \(n\), and \(n=1\) corresponds to 2001 . (a) Find the partial sums \(S_{3}, S_{4}, S_{5}, S_{6}\). (b) Find \(S_{5}-S_{4}, S_{6}-S_{5}\) (c) Use your answer to part (b) to explain the value of \(S_{n+1}-S_{n}\) for any positive integer \(n .\) $$ \begin{array}{l|c|c|c|c|c|c} \hline \text { Year } & 2001 & 2002 & 2003 & 2004 & 2005 & 2006 \\ \hline n & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline \text { Deaths } & 17,402 & 16,948 & 16,690 & 16,395 & 16,268 & 14,016 \\\ \hline \end{array} $$

Find the sum of the series. $$ 1 / 81+1 / 27+1 / 9+\cdots+243 $$
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