P1 Chapter 14 CIE Centre A-level Pure Maths © Adam Gibson

The first is an arithmetic sequence. If we add the terms together, we get a sum sequence or a series. We know the formula for this type of sequence already (Chapter 8). But what about the second type? The sum of the terms in B is usually called a series sequence progression A B SEQUENCES AND SERIES

Proof of the finite formula For a finite geometric sequence or series, such as: We proceed as follows. In this series, 3 is called the common ratio, because of the equivalent inductive definition: Write the first 4 terms of the sequence See the box on p. 210 – definition of a geometric sequence/progression To find the value of the series (the sum), just multiply it by the common ratio:

Proof of the finite formula – continued. Do you see the trick? Look: The two series are the same, except for the first and last term. And obviously the difference between them is 2S.

Proof of the finite formula – continued. Notice – they can become big very fast! It is easy then to understand the general formula: If the common ratio is r, and the first term is a, and the number of terms is n, the sum is found thus: “It is more important to understand than to remember”

Extending the formula What happens if r is negative? A: Nothing. The formula is still correct. Here is an example: What are a, n and r? a=4, n=7, r=-1/2

Extending the formula – infinite series. What happens if n is infinite?A: It depends on r.

Infinite geometric series The above graphs are easy to understand in terms of the finite formula: If |r| < 1, we say that the series is convergent. If not, we say that the series is divergent. (note from the graph that there are two different “kinds” of divergence). For the infinite sum, we write

A wordy example… Meera invests $2,000 in a building society account on 1 January 2000 and the same amount on 1 Jan each succeeding year. If the building society pays compound interest at 4.5% per annum, calculate how much is in Meera’s account on 31 December Answer: a= 2000*1.045 n= 11 r= So S=$28, to 2d.p. (to the nearest cent).

Practice Tasks from Chapter 14 Remember the key formulae: p. 213 Q1 d, Q4 b,d Q5a,d,j Q10 p. 217 Q1 a,d,e Q2 a,d Q5 Q10 p. 221 Q1,3,8 Misc Exercise Q4, Q7, Q16, Q19, Q21 (hard)