Geometric Progressions

This ppt is only based on Geometric Progression with some examples of it.

1. GEOMETRIC PROGRESSION

2. What is it? In Mathematics, a geometric progression, also known as a geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. It is denoted by the letter “r”. In a G.P, the first term of a sequence is denoted by the letter “a”

3. Introduction A   geometric sequence   is a sequence such that any element after the first is obtained by multiplying the preceding element by a constant called the   common ratio   which is denoted by r. The common ratio (r) is obtained by dividing any term by the preceding term, i.e., The  geometric sequence  is sometimes called the  geometric progression  or  GP , for short.

4. Let’s take a look at the formulae for finding the n- th term of a G.P If the first term is denoted by a , and the common ratio by r , the series can be written as: a + e.g. 5 + 10 + 20 + 40 + … Hence the n th term is given by: or 2 – 4 + 8 –16 + … ar 2 + ar + ar 3 + … Geometric Progressions

5. Geometric Progressions The sum of the first n terms, S n is found as follows: S n = a + ar + ar 2 + ar 3 +… ar n –2 + ar n –1 …(1) Multiply throughout by r : r S n = ar + ar 2 + ar 3 + ar 4 + … ar n –1 + ar n …(2) Now subtract (2) – (1): r S n – S n = ar n – a Factorise: S n ( r – 1) = a ( r n – 1 ) Hence:

6. Elementary Properties The behaviour of a geometric sequence depends on the value of the common ratio. If the common ratio is: Positive, the terms will all be the same sign as the initial term. Negative, the terms will alternate between positive and negative. Greater than 1, there will be  exponential growth  towards  positive or negative infinity  (depending on the sign of the initial term). 1, the progression is a  constant  sequence. Between −1 and 1 but not zero, there will be  exponential decay  towards zero. −1, the progression is an alternating sequence Less than −1, for the absolute values there is exponential growth towards  (unsigned) infinity , due to the alternating sign.  

7. Selecting terms in a G.P A GP of 3 terms : a/r, a, ar A GP of 4 terms : a/r 3 , a/r, ar , ar 3 And so on. If all terms in a GP are multiplied or divided by the same number, or are raised to the same power, then the resulting sequence is still a GP.  

8. Geometric Mean If a, b, c are in GP then b 2 =ac and b is called the GM of a and c. Conversely, if b 2 =ac, then a, b, c are in GP. Sum of infinite terms of a GP:  If -1<r<1, then GP is said to converge, that is to say that sum of infinite terms of such a GP tends to a constant value.   

9. Examples Example 1 : For the series 2 + 6 + 18 + 54 + … Find a) The 10 th term. b) The sum of the first 8 terms. a) For the series, we have: a = 2, r = 3 Using: u n = ar n –1 u 10 = 2(3 9 ) = 39 366 = 6560

10. Examples Example 2 : For the series 32 – 16 + 8 – 4 + 2 … Find a) The 12 th term. b) The sum of the first 7 terms. a) For the series, we have: a = 32, r = Using: u n = ar n –1 u 12 = 32 = 21.5 1 2 – ( ) 1 2 – 11 = 1 64 – We can write this as:

11. Examples Example 3 : In a Geometric Series, the third term is 36, and the sixth term is 121.5. For the series, find the common ratio, the first term and the twentieth term. The third term is 36 i.e. u 3 = 36 The sixth term is 121.5 i.e. u 6 = 121.5 Using: u n = ar n –1 ar 2 = 36 ….(1) ar 5 = 121.5 ….(2) Now, divide equation (2) by equation (1): So r 3 = 3.375 r = 1.5 Substitute this value into equation (1): a (1.5) 2 = 36 a = 16 Now the 20 th term, u 20 = ar 19 = 16 (1.5) 19 = 35 469 (To the nearest integer)

12. Examples Example 4 : The first three terms of a geometric progression are x , x + 3, 4 x . Find the two possible values for the common ratio. For each value find these first three terms, and the common ratio. The ratio of a G.P. is found by dividing a term by the previous term: Now, x (4 x ) = ( x + 3)( x + 3) 3 x 2 – 6 x – 9 = 0 Divide by 3: x 2 – 2 x – 3 = 0 ( x – 3)( x + 1) = 0 So, either x = 3, giving the terms: 3, 6, 12 with ratio r = 2 or x = –1, giving the terms: –1, 2, –4 with ratio r = –2 4 x 2 = x 2 + 6 x + 9

13. Summary A Geometric Series is one in which the terms are found by multiplying each term by a fixed number (common ratio). The n th term is given by: The sum of the first n terms is given by: In problems where r < 1 it is better to write the above as:

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