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SL 1.3 Geometric sequences and series Study Notes-  IBDP Maths AA- SL & HL – New Curriculum 2021-2028

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Topics: SL 1.3 Geometric sequences and series Study Notes- IBDP Maths AA- SL & HL  based on New Curriculum 2021-2028

GEOMETRIC SEQUENCE….

• THE DEFINATION..

I give you the first term of a sequence, say $$u_1$$ =5 and  I ask you to multiply by a fixed number, say r =2, in order to find the next term. The following sequence is generated:

5, 10, 20, 40, 80, …

Such a sequence is called geometric. That is, in a geometric sequence the ratio between any two consecutive terms is constant.

In other words , we can say , geometric sequence is a sequence where there is a common ratio between consecutive number.

For determining geometric sequence , we only need

• The first term
• The common ratio
• Common ratio:  The common ratio is the constant factor by which each term in a geometric sequence is multiplied to get the next term.
 EXAMPLE 1(a) $$u_1$$ =1,  r =2        the sequence is 1, 2, 4, 8, 16, 32, 64, …(b) $$u_1$$ =5,  r = 10    the sequence is 5, 50, 500, 5000, …(c) $$u_1$$ =1,  r = -2      the sequence is 1,-2, 4,-8, 16, …(d) $$u_1$$ =1, r = $$\frac{1}{2}$$    the sequence is  1, $$\frac{1}{2}$$, $$\frac{1}{4}$$, $$\frac{1}{16}$$, $$\frac{1}{32}$$, …..(e) $$u_1$$ =1, r = $$\frac{-1}{2}$$    the sequence is 1, $$\frac{-1}{2}$$, $$\frac{1}{4}$$, $$\frac{1}{16}$$, $$\frac{-1}{32}$$, …..

ATTENTION!!

• The common ratio r may also be negative! In this case the signs alternate (+, -, +, -, …) [see (c) and (e) above].
• The common ratio r may be between -1 and 1, that is |r|<1. In such a sequence the terms approach 0 [see (d) and (e) above]
• THE $$N^{th}$$ TERM FORMULA

The nth term formula for a geometric sequence is $$u_n =u_1r^{n-1}$$

where,

• $$u_1$$ is the first term
• n is the term number
• r is the common ratio

Indeed, let us think:
In order to find $$u_5$$ , we start from u1 and then multiply 4 times by the ratio r

Hence, $$u_5 = u_1r^4$$

Similarly, $$u_{10} = u_1r^9$$, $$u_{50} = u_1r^{49}$$ and so on…..

 EXAMPLE 2In a geometric sequence let $$u_1$$ =3 and r =2. Find (a) the first four terms(b) the 100th termSolution (a) 3, 6, 12, 24 (b) Now we need the general formula$$u_{100} = u_1r^{99} = 3.2^{99}$$EXAMPLE 3 In a geometric sequence let $$u_1$$ =10 and $$u_{10} = 196830 . Find \(u_3$$Solution We know $$u_1$$ , we need r. We exploit the information for $$u_{10}$$ first.$$u_{10}=u_1r^9 = 196830 = 10 . r^9$$$$r^9= 19683$$$$r= \sqrt[9]{19683} = 3$$Therefore , $$u_3 = u_1r^2 = 10.3^2 = 90$$

REMEMBER!!

Our first task in a G.S. is to find the basic elements, $$u_1$$ and r , and then anything else.

• SUM OF N TERMS($$S_n$$

Given that r ≠ 1 , the result is given by

$$S_n = \frac{u_1(r^n-1)}{r-1}$$ or  $$S_n = \frac{u_1(1-r^n)}{1-r}$$

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