Question

A recursive arithmetic sequence is defined ad 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. Find the slope of the line that passes through the points given in the table. The slope is

Answer 1

Write the first few terms of the sequence.
f₁ = 6
f₂ = f₁ + 5 = 11
f₃ = f₂ + 5 = 16
f₄ = f₃ + 5 = 21
and so on

The sequence is
6, 11, 16, 21, ...,
This is an arithmetic sequence with a common difference of 5.

In function notation, the sequence is
$f_{n+1} = 6 + 5n, \\ \, n=0,1,2, \, ...,$

This represents the equation of a straight line with n as the independent variable and f as the dependent variable.
The slope is 5. 
A graph of the line is shown below.

Answer:
The sequence is $f_{n+1} = 6 + 5n, \,\, n=0,1,2, \, ...,$
The slope is 5.

Answer 2

Answer:

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.

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