A bus pass has a starting value of $100. After one ride, the value of the pass is $98.25. After two rides, its value is $96.50. After three rides, its value is $94.75.
Write a rule to represent the remaining value on the card as an arithmetic sequence.
what i know: i know its $1.75 per bus pass, I don't know how to write a arithmetic sequence and i need help if someone would please explain it to me how
Let x = number of rides
100 - 1.75x = ?
To represent the remaining value on the bus pass as an arithmetic sequence, we need to express the values as a series of terms that follow a regular pattern based on the number of rides.
In this case, the starting value of $100 decreases by $1.75 after each ride.
Let's break down the given information:
Starting value: $100
Value after one ride: $98.25
Value after two rides: $96.50
Value after three rides: $94.75
To find the common difference (the amount by which the value decreases after each ride), we subtract the value of the previous term from the value of the current term.
Common difference = $98.25 - $100 = -$1.75, or -1.75
Thus, the rule to represent the remaining value on the card as an arithmetic sequence is:
Value after n rides = $100 - ($1.75 * (n-1))
or
Value after n rides = $100 - $1.75n + $1.75, where n represents the number of rides taken.
To represent the remaining value on the bus pass as an arithmetic sequence, we can start by identifying the common difference, which is the amount deducted from the card after each ride.
In this case, we can see that the value of the pass decreases by $1.75 after each ride. So, the common difference is -1.75 (negative because the value is decreasing).
Now, let's express the remaining values on the card after each ride using the arithmetic sequence. We can start with the initial value of $100 and subtract the common difference (-1.75) after each ride.
The first term of the sequence (n=1) would be $100 since that's the initial value. To find the remaining values, we can use the formula:
an = a1 + (n - 1)d
Where:
an = value after n rides
a1 = initial value
n = number of rides
d = common difference
Substituting the values we have:
a2 = $100 + (2 - 1)(-1.75)
a3 = $100 + (3 - 1)(-1.75)
and so on.
Simplifying:
a2 = $100 - $1.75 = $98.25
a3 = $98.25 - $1.75 = $96.50
Therefore, the rule representing the remaining value on the bus pass as an arithmetic sequence is:
$100, $98.25, $96.50, $94.75, ...
In general, to find the value after n rides, use the formula:
an = $100 + (n - 1)(-1.75)