Question

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

Short answer

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

Step-by-step solution

Function of the sequence given

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

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: $$ a_1 = 2 $$

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: $$ a_2 = 5 $$

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: $$ a_3 = 8 $$

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: $$ a_4 = 11 $$

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: $$ a_5 = 14 $$

Writing down the first 5 terms

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

Key concepts

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.