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.
Wayne and Dennis like to ride the bike path from Riverside Park to the beach. Dennis’s speed is seven miles per hour faster than Wayne’s speed, so it takes Wayne 2 hours to ride to the beach while it takes Dennis 1.5 hours for the ride. Find the speed of both bikers.
from theory: distance [miles] = speed [mph] × time [hours]
info #1
speed_Dennis × 1.5 = speed_Wayne × 2
=> speed_Wayne = 0.75 × speed_Dennis (i)
info #2
speed_Dennis = speed_Wayne + 7 [mph] (ii)
use (i) in (ii) => [...]
speed_Dennis = 28 mph
speed_Wayne = 21 mph
George
Let W be Wayne's speed in miles per hour and D be Dennis's speed in miles per hour. We know that W + 7 = D and W * 2 = D * 1.5.
Substituting the first equation into the second:
W * 2 = (W + 7) * 1.5
W * 2 = W * 1.5 + 7 * 1.5
0.5 * W = 7 * 1.5
W = 7 * 3 or 21
W is 21
D = W + 7
D = 21 + 7
D = 28
Salma
Devon is 32 32 years older than his son, Milan. The sum of both their ages is 54 54. Using the variables d d and m m to represent the ages of Devon and Milan, respectively, write a system of equations to describe this situation. Enter the equations below, separated by a comma.
please why is it that the 0is in the place of ten thousand
Grace
Send the example to me here and let me see
Stephen
A meditation garden is in the shape of a right triangle, with one leg 7 feet. The length of the hypotenuse is one more than the length of one of the other legs. Find the lengths of the hypotenuse and the other leg
however, may I ask you some questions about Algarba?
Amoon
hi
Enock
what the last part of the problem mean?
Roger
The Jones family took a 15 mile canoe ride down the Indian River in three hours. After lunch, the return trip back up the river took five hours. Find the rate, in mph, of the canoe in still water and the rate of the current.
Shakir works at a computer store. His weekly pay will be either a fixed amount, $925, or $500 plus 12% of his total sales. How much should his total sales be for his variable pay option to exceed the fixed amount of $925.
I'm guessing, but it's somewhere around $4335.00 I think
Lewis
12% of sales will need to exceed 925 - 500, or 425 to exceed fixed amount option. What amount of sales does that equal? 425 ÷ (12÷100) = 3541.67. So the answer is sales greater than 3541.67.
Check:
Sales = 3542
Commission 12%=425.04
Pay = 500 + 425.04 = 925.04.
925.04 > 925.00
Munster
difference between rational and irrational numbers
Jazmine trained for 3 hours on Saturday. She ran 8 miles and then biked 24 miles. Her biking speed is 4 mph faster than her running speed. What is her running speed?