A recursive arithmetic sequence is defined as

Question

𝑓
(
1
)
=
6
,
𝑓
(
𝑛
+
1
)
=
𝑓
(
𝑛
)
+
5
for
𝑛
≥
1
. The first four terms of the sequence are shown in the table. Write an explicit formula that represents the sequence using function notation.

A two column table with 5 rows. The first row, n, has the entries, 1, 2, 3, 4. The second column, f(n), has the entries, 6, 11, 16, 21. Complete the steps to write an explicit formula that represents the sequence.

Find the slope of the line that passes through the points given in the table. The slope is .

Use one of the given points to find the y-intercept. Substitute values for x, y, and m into the equation
𝑦
=
𝑚
𝑥
+
𝑏
and solve for b. The y-intercept is .

Write the formula as a function of n in slope-intercept form. The function is for n in the set of natural numbers.

Answer 1

To derive the explicit formula for the recursive arithmetic sequence defined by \( f(1) = 6 \) and \( f(n + 1) = f(n) + 5 \) for \( n \geq 1 \), we can follow the steps given:

Step 1: Find the Slope

From the terms given in the sequence:

• \( f(1) = 6 \)
• \( f(2) = 11 \)
• \( f(3) = 16 \)
• \( f(4) = 21 \)

The difference between consecutive terms can be calculated as follows:

• \( f(2) - f(1) = 11 - 6 = 5 \)
• \( f(3) - f(2) = 16 - 11 = 5 \)
• \( f(4) - f(3) = 21 - 16 = 5 \)

The common difference (slope) is 5. Thus, the slope \( m \) is: \[ m = 5 \]

Step 2: Find the Y-Intercept

We can use one of the points from the table, for example \( (1, 6) \), to find the y-intercept \( b \). The slope-intercept form of the equation is: \[ y = mx + b \]

Substituting \( y = f(n) \), \( m = 5 \), and the point \( (x, y) = (1, 6) \): \[ 6 = 5(1) + b \] \[ 6 = 5 + b \] To find \( b \): \[ b = 6 - 5 = 1 \]

Step 3: Write the Formula as a Function of n

Now, we can write the equation in slope-intercept form: \[ f(n) = 5n + 1 \]

Conclusion

Thus, the explicit formula that represents the sequence \( f(n) \) is: \[ f(n) = 5n + 1 \quad \text{for } n \in \mathbb{N} \]

This completes our formulation for the recursive arithmetic sequence.

Answer 2

Find the slope of the line that passes through the points given in the table. The slope is
✔ 5
.
Use one of the given points to find the y-intercept. Substitute values for x, y, and m into the equation y = mx + b and solve for b. The y-intercept is
✔ 1
.
Write the formula as a function of n in slope-intercept form. The function is
✔ f(n) = 5n+ 1
for n in the set of natural numbers.

Which functions represent the arithmetic sequence 8, 1.5, –5, –11.5 . . . ? Check all that apply.

f(n) = –6.5n + 14.5
f(n) = –1.5n + 9.5
f(n) = 6.5n + 1.5
f(1) = 8, f(n + 1) = f(n) – 6.5
f(1) = 8, f(n + 1) = f(n) – 1.5
f(1) = 8, f(n + 1) = f(n) + 6.5

Answer 3

Let's analyze the given arithmetic sequence \( 8, 1.5, -5, -11.5, \ldots \) to determine which functions represent it.

Step 1: Identify the Common Difference

To find the common difference, we subtract each term from the subsequent term:

• \( 1.5 - 8 = -6.5 \)
• \( -5 - 1.5 = -6.5 \)
• \( -11.5 - (-5) = -6.5 \)

The common difference is \( -6.5 \).

Step 2: Write the General Form of the Sequence

An arithmetic sequence can be expressed in the form \( f(n) = f(1) + (n - 1)d \), where \( d \) is the common difference.

1. The first term is \( f(1) = 8 \).
2. The common difference is \( d = -6.5 \).

Thus, the explicit formula for the sequence can be written as: \[ f(n) = 8 + (n-1)(-6.5) \] Simplifying further: \[ f(n) = 8 - 6.5(n - 1) = 8 - 6.5n + 6.5 = -6.5n + 14.5 \]

Step 3: Compare with Given Functions

Now we check each of the given functions to see which represent the sequence:

1. \( f(n) = -6.5n + 14.5 \)

   • This matches our derived formula. ✔

2. \( f(n) = -1.5n + 9.5 \)

   • This does not match. ❌

3. \( f(n) = 6.5n + 1.5 \)

   • This does not match. ❌

4. \( f(1) = 8, f(n + 1) = f(n) - 6.5 \)

   • This is consistent with the series: the next term is obtained by subtracting 6.5, which matches the common difference. ✔

5. \( f(1) = 8, f(n + 1) = f(n) - 1.5 \)

   • This does not match as the common difference is \( -6.5 \). ❌

6. \( f(1) = 8, f(n + 1) = f(n) + 6.5 \)

   • This also does not match since we are adding instead of subtracting. ❌

Conclusion

The functions that represent the arithmetic sequence \( 8, 1.5, -5, -11.5, \ldots \) are:

• \( f(n) = -6.5n + 14.5 \)
• \( f(1) = 8, f(n + 1) = f(n) - 6.5 \)

Thus, the correct options are checked as follows:

• ✔ \( f(n) = -6.5n + 14.5 \)
• ✔ \( f(1) = 8, f(n + 1) = f(n) - 6.5 \)

Answer 4

f(1) = 8, f(n + 1) = f(n) – 6.5
f(n) = –6.5n + 14.5
only right

Answer 5

Apologies for any confusion in my previous response. Let’s clarify which functions correctly represent the arithmetic sequence \( 8, 1.5, -5, -11.5, \ldots \).

Given Sequence

The sequence is: \[ 8, 1.5, -5, -11.5, \ldots \] The common difference \( d \) is \( -6.5 \).

Functions to Check

1. \( f(1) = 8, f(n + 1) = f(n) - 6.5 \)

   • This accurately represents the sequence, as it states that each term is obtained by subtracting \( 6.5 \) from the previous term. ✔️

2. \( f(n) = -6.5n + 14.5 \)

   • This is the explicit formula we derived, which gives the correct value for the terms in the sequence. When \( n = 1 \): \( f(1) = -6.5(1) + 14.5 = -6.5 + 14.5 = 8 \), and so forth for subsequent values of \( n \). ✔️

Functions That Do Not Represent the Sequence

3. \( f(n) = -1.5n + 9.5 \)

   • Does not represent the sequence since the slope does not match \( -6.5 \). ❌

4. \( f(n) = 6.5n + 1.5 \)

   • Also incorrect, as it suggests an increasing sequence rather than a decreasing one. ❌

5. \( f(1) = 8, f(n + 1) = f(n) - 1.5 \)

   • Incorrect, because it suggests subtracting \( 1.5 \) instead of \( 6.5 \). ❌

6. \( f(1) = 8, f(n + 1) = f(n) + 6.5 \)

   • Incorrect, as it implies the function is increasing by \( 6.5 \). ❌

Summary

The only correct options for the given arithmetic sequence are:

• \( f(1) = 8, f(n + 1) = f(n) - 6.5 \)
• \( f(n) = -6.5n + 14.5 \)

You are correct that these two functions accurately represent the arithmetic sequence. Thank you for your patience.