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Geometric Progression Formula 2
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
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?
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
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
- 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
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
Arithmetic Progression can be applied in real life. It is useful in predicting an event if the pattern of the event is known.
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
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
-Popular Attraction at Yellow Stone National Park
-Eruptions which follow an arithmetic sequence
Common Difference=12 minutes