arithmetic & geometric sequence and series

Arithmetic Sequence

Situation

While some sequences are simply random values,

other sequences have a definite pattern that is used to arrive at the sequence's terms.

Two such sequences are the arithmetic and geometric sequences.

Your room is too cold, so you decide to adjust the thermostat. The current temperature of the room is 60˚ Fahrenheit. In an attempt to get warmer, you increase the temperature to 62˚ . When this doesn't warm the room enough for you, you decide to increase the thermostat to 64˚ . This temperature still isn't warm enough, so you continue to increase it in this manner.

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Applications of Geometric Series in “Real Life”

An Geometric Sequence describes something that is periodically growing in an exponential fashion (by the same percentage each time), and a Geometric Series describes the sum of those periodic values. Examples of Geometric Series that could be encountered in the “real world” include:

– What is the total number of births over a 20 year period to a population that grows at a fixed percentage each year?

– How much interest will $1,000 invested in a fixed-rate certificate of deposit earn over 30 years?

– How much of a medicine that a patient takes every 8 hours remains in the body after taking it for 48 hours if only 20% remains after each 8 hour period?

To find any term of a geometric sequence:

an = ar(n-1)

Applications of Geometric Sequences in “Real Life”

To sum:

a + ar + ar2 + ... + ar(n-1)

They can model situations that involve a constant rate of growth, but where the only inputs that make sense are integers. Examples include:

– Annual size of a population that is growing (or shrinking) at a constant rate

– Value of money in an account that receives periodic fixed rate interest payments

– Maximum height of a bouncing ball after each bounce (if a fixed % of its energy is lost on each bounce)

– Radioactivity levels of a sample over time

To find the sum of a finite geometric series, use the formula,

Sn=a1(1−r^n)/1−r,r≠1 or

Sn=a1/1−r

Geometric Sequence & Series

-If a sequence of values follows a pattern of multiplying a fixed amount (not zero) times each term to arrive at the following term, it is referred to as a geometric sequence.

-The fixed amount multiplied is called the common ratio, r. To find the common ratio, divide the second term by the first term.

-A geometric series is a series whose related sequence is geometric.It results from adding the terms of a geometric sequence.

A sequence is an ordered list of numbers.

The sum of the terms of a sequence is called a series.

Questions:

Find the common ratio for the sequence

6,-3,3/2,-3/4,....

r= multiplying each entry by -1/2

The third term of a geometric sequence is 3 and the sixth term is 1/9. Find the first term.

a1 = 27

A ball is dropped from a height of 8 feet. The ball bounces to 80% of its previous height with each bounce. How high (to the nearest tenth of a foot) does the ball bounce on the fifth bounce?

an (a5)=2.6 feet

Visual Representation

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While it may be difficult to see in the images provided, the temperature is being increased by 2 F each time. This created the arithmetic sequence of 60, 62, 64, 68, 70.

Arithmetic and Geometric Sequence and Series

Ans.

The seating pattern is forming an arithmetic sequence.

60, 68, 76, ...

We wish to find "the sum" of all of the seats.

n = 20, a1 = 60, d = 8 and we need a20 for the sum.

Now, use the sum formula:

There are 2720 seats.

Q.

A theater has 60 seats in the first row, 68 seats in the second row, 76 seats in the third row, and so on in the same increasing pattern. If the theater has 20 rows of seats, how many seats are in the theater?

Examples

1, 4, 7, 10, 13, 16, ... d = 3

add 3 to each term to arrive at the next term,or...the difference a2 - a1 is 3.

15, 10, 5, 0, -5, -10, ... d = -5

add -5 to each term to arrive at the next term, or...the difference a2 - a1 is -5.

Arithmetic sequence

-If a sequence of values follows a pattern of adding a fixed amount from one term to the next, it is referred to as an arithmetic sequence. The number added to each term is constant (always the same).

-The fixed amount is called the common difference, d, referring to the fact that the difference between two successive terms yields the constant value that was added.

-To find the common difference, subtract the first term from the second term.

Formulas used with arithmetic sequences and arithmetic series:

Thank you

To find the sum of a certain number of terms of an arithmetic sequence:

where Sn is the sum of n terms (nth partial sum),

a1 is the first term, an is the nth term.

To find any term

of an arithmetic sequence:

where a1 is the first term of the sequence,

d is the common difference, n is the number of the term to find.

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Mouza, Shamma, Fatima Mخ

Have you ever been in math class and wondered ... when will I ever use math in real life?