Geometric Progression

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Topic : Geometric Progression

First of all, what is mean by Geometric Progression? Geometric Progression is a sequence of numbers in which each term is formed by multiplying the previous term by the same number or expression.

Formula for Geometric Progression :- 

Tn = ar(n-1)

r    = U3 = U2 = U4 = ................
         U2    U1    U3

S (infinity) = a
                     1-r

Here's some of the example of the Geometric Progression.

Example 1

Given the following numbers :-

3,-16,12,-2,........

a) Calculate the 27th term of the sequence.

Note
The question ask to calculate the 27th term of the sequence.

So, the first things you must do is find the difference between the two number.

Step 1

-6  = -2  , 12 = -2  
3             -16  

So r = -6      
           3
        = -2

We use formula of Tn = ar(n-1) to calculate the 27th term of the sequence.

Tn = ar(n-1)

Step 2

We identify first.

n = 27

a = 3

r = -2

Step 3

We include the number (number after we identify) into the formula.

Tn = ar(n-1)

T23 = 3 (-2)' 27-1

        = 3 (-2)' 26

        = 201326592

So, the 27th term of the sequence is 201326592.

Example 2

Given the following numbers :-

3,-16,12,-2,........

a) Find the sum of the series until the 32nd term.

Okay~ The question ask us to find the sum of the series until the 32nd term.

So, the first thing we must do is choesses the formula first. The question ask to find the sum of the series until the 32nd term. So that mean we use sum formula.

Sn = a (1-r'n ) 
            1-r

Step 1

Identify the number first.

n = 32

a = 3

r = -2

Step 2

Include the number into the formula.

Sn = a (1-r'n ) 
            1-r

Step 2

Input and calculate the number using formula to find the answer.

Sn = a (1-r'n ) 
            1-r

S32 = 3 (1-(-2)' 32)
               1-(-2)

Step 3

 S32 = 3 (1-(-2)' 32)  .............................. (- x - = + )
               1-(-2)          .............................. (- x - = + )

S32 = 3 (1 + 2)' 32)
                 1 + 2)
S32 = 3 [ 1 - 4294967296 ]
                       3

S32 = -4.294,967,296

So, the answer for the sum of the series until 32nd term is -4.294,967,296