Solving application problems with arithmetic sequences
In many application problems, it often makes sense to use an initial term of
instead of
In these problems, we alter the explicit formula slightly to account for the difference in initial terms. We use the following formula:
Solving application problems with arithmetic sequences
A five-year old child receives an allowance of $1 each week. His parents promise him an annual increase of $2 per week.
Write a formula for the child’s weekly allowance in a given year.
What will the child’s allowance be when he is 16 years old?
The situation can be modeled by an arithmetic sequence with an initial term of 1 and a common difference of 2.
Let
be the amount of the allowance and
be the number of years after age 5. Using the altered explicit formula for an arithmetic sequence we get:
We can find the number of years since age 5 by subtracting.
We are looking for the child’s allowance after 11 years. Substitute 11 into the formula to find the child’s allowance at age 16.
The child’s allowance at age 16 will be $23 per week.
A woman decides to go for a 10-minute run every day this week and plans to increase the time of her daily run by 4 minutes each week. Write a formula for the time of her run after n weeks. How long will her daily run be 8 weeks from today?
recursive formula for nth term of an arithmetic sequence
explicit formula for nth term of an arithmetic sequence
Key concepts
An arithmetic sequence is a sequence where the difference between any two consecutive terms is a constant.
The constant between two consecutive terms is called the common difference.
The common difference is the number added to any one term of an arithmetic sequence that generates the subsequent term. See
[link] .
The terms of an arithmetic sequence can be found by beginning with the initial term and adding the common difference repeatedly. See
[link] and
[link] .
A recursive formula for an arithmetic sequence with common difference
is given by
See
[link] .
As with any recursive formula, the initial term of the sequence must be given.
An explicit formula for an arithmetic sequence with common difference
is given by
See
[link] .
An explicit formula can be used to find the number of terms in a sequence. See
[link] .
In application problems, we sometimes alter the explicit formula slightly to
See
[link] .
Section exercises
Verbal
What is an arithmetic sequence?
A sequence where each successive term of the sequence increases (or decreases) by a constant value.
Step 1: Find the mean. To find the mean, add up all the scores, then divide them by the number of scores. ...
Step 2: Find each score's deviation from the mean. ...
Step 3: Square each deviation from the mean. ...
Step 4: Find the sum of squares. ...
Step 5: Divide the sum of squares by n – 1 or N.
The sample of 16 students is taken. The average age in the sample was 22 years with astandard deviation of 6 years. Construct a 95% confidence interval for the age of the population.
Bhartdarshan' is an internet-based travel agency wherein customer can see videos of the cities they plant to visit. The number of hits daily is a normally distributed random variable with a mean of 10,000 and a standard deviation of 2,400
a. what is the probability of getting more than 12,000 hits?
b. what is the probability of getting fewer than 9,000 hits?
Bhartdarshan'is an internet-based travel agency wherein customer can see videos of the cities they plan to visit. The number of hits daily is a normally distributed random variable with a mean of 10,000 and a standard deviation of 2,400.
a. What is the probability of getting more than 12,000 hits