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Arithmetic Progression and Geometric Progression

Arithmetic Progression Formula 2

Arithmetic Progression

Extension

Geometric Progression Formula 2

Geometric Progression

To find the sum of n terms

Sn=n/2 (a+l), where a is the first term of the series and l denotes the last term of the series of n terms.

Geometric Progression can be applied to daily life

For eg. Calculate bank interests

An arithmetic progression is a sequence in which the term that comes after the term before it always has the same difference.

For eg. 1,3,5,7,9,11

A geometric progression is a sequence in which each term after the first is found by multiplying the previous term with a fixed number (the common ratio)

Sum of geometric series:

Sn=a(1-r^n)/1-r

where a is the first

term and r is the common ratio

The Bank Interest Problem

Vanessa deposited some money into her bank account at the beginning of 2010 which is compounded annually. At the end of 2010, the money will be $200. The money will be $210 and $220.15 respectively for the subsequent years. How much will she have at the beginning of the 5th year?

Example 2:

Solution

Step 1: Establish that a=200, r=1.05

T4= 200(1.05)^(4-1)

=200(1.157625)

=231.53

For eg. 2, 1, 0.5, 0.25

Find the sum of the series

2,4,6,8,10

Step 1: Establish a=2, l=10,

Sn= 5/2(10+2)

Sn= 5/2(12)

Sn=30

Reflections

Example 5

Find the sum of the geometric series where there are 6 terms in the series

2,6,18,54,.....

Step 1: We can deduce that a=2, r=3 and n=6

Sn=a(1-r^n)/1-r

=2(1-3^6)/1-3

=2(1-729)/-2

=-(-728)

=728

Arithmetic Progression Formula 1

Arithmetic Progression Formula 3

- Allow us to delve deeper into this concept which we have briefly touched on in lower sec

-Geometric Progression is an eye opener as we can apply the formulas in real life

-Learning about the progressions is not an easy process

Geometric Progression Formula 1

To find the nth term

Tn=a+(n-1)d , where a is the first

term of the series and d is the common difference

To find the sum of n terms

Sn=n/2(2a+(n-1)d) where a is the first terms and d is the common difference

Extension

To find the nth term,

Tn=ar^(n-1), where a is the first term of the series and r is the common ratio

Example 1:

Arithmetic Progression can be applied in real life. It is useful in predicting an event if the pattern of the event is known.

Example 3

2,6,10,14,18,x

Find the sum of the first 50 terms of the sequence

1,3,5,7,9.....

Step 1: establish that a=1, d=2 and n=50

Sn=n/2(2a+(n-1)d)

S50=1/2x50x(2x1+(49)2)

Sum=2500

Step 1: Establish that x is the 6th term

Step 2: a is the first term aka 2

Step 3: d is 18-14=4 OR 10-6=4

Step 4: Apply the formula

T6=a+(n-1)d

=2+(6-1)4

=22

Example 4:

Write down the first five terms of the

geometric progression which has first

term 1 and common ratio 1/2.

T2=1(1/2)^(2-1)

=1/2

T3=1(1/2)^(3-1)

=1/4

T4= 1(1/2)^(4-1)

=1/8

T5= 1(1/2)^(5-1)

=1/16

Ans: 1,1/2,1/4,1/8,1/16

Old Faithful Geezer

-Popular Attraction at Yellow Stone National Park

-Eruptions which follow an arithmetic sequence

Common Difference=12 minutes

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