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.
Astronomy (from Ancient Greek ἀστρονομία (astronomía) 'science that studies the laws of the stars') is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution.
Rafael
vjuvu
Elgoog
what is big bang theory?
Rosemary
what type of activity astronomer do?
Rosemary
No
Richard
the big bang theory is a theory which states that all matter was compressed together in one place the matter got so unstable it exploded releasing All its contents in the form of hydrogen
according to the theory of astronomers why the moon is always appear in an elliptical orbit?
Gatjuol
hi !!! I am new in astronomy....
I have so many questions in mind ....
all of scientists of the word they just give opinion only.
but they never think true or false ...
i respect all of them...
I believes whole universe depending
on true ...থিউরি
Govinda
hello
Jackson
hi
Elyana
we're all stars and galaxies a part of sun. how can science prove thx with respect old ancient times picture or books..or anything with respect to present time .but we r a part of that universe
there many theory to born universe but what is the reality of big bang theory to born universe
Asmit
what is the exact value of π?
Nagalakshmi
by big bang
universal
there are many theories regarding this it's on you believe any theory that you think is true ex. eternal inflation theory, oscillation model theory, multiple universe theory the big bang theory etc.
Aarya
I think after Big Bang!
Michele
from where on earth could u observe all the stars during the during the course of an year
is that so. the question was in the end of this chapter
Karuna
in theory, you could see them all from the equator (though over the course of a year, not at pne time). stars are measured in "declination", which is how far N or S of the equator (90* to -90*). Polaris is the North star, and is ALMOST 90* (+89*).
So it would just barely creep over the horizon.
Christopher
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