1, 2, 3, 4, 5, 19, 7, 8, 12
20, 18, 16, 14, 12, 10, 8
1, 0.1, 0.01, 0.001, 0.0001, 0.00001, 0.000001
The individual numbers or members of the sequence are called the terms of the sequence. Usually the sequence is created with some rhyme or reason, such as the last two above. The second one was created by using the formula 22 - 2*N for N = 1 to 7, while the last used 10/10N for N = 1 to 7. Your computer is good at creating sequences. The following simple program illustrates this. Insert your own formula (involving N) in the second line.
Listing 5.1
LIST
10 REM equence generator 20 MODE 1:COLOUR 3:PRINT ' TAB(11);"S equence generator"':@%=10 30 PRINT "This program produces seque nces."' 40 PRINT "Enter your formula involvin g N, for example"' 50 PRINT " (N*N-N+4/2"' 60 INPUT "Formula involving N: ";N$ 70 DEF FNFormula(N)=EVAL(N$) 80 COLOUR 1:PRINT '"Here are the firs t ten terms:"':COLOUR 2 90 FOR N=1 TO 10 100 IF 38-POS<LEN(STR$(FNFormula(N))) THEN PRINT 110 PRINT ;FNFormula(N);:IF N<10 THEN PRINT ", "; 120 NEXT 130 PRINT 140 COLOUR 3:PRINT CHR$(7) ' TAB(10);" Another go? Y or N "; 150 REPEAT:G$=GET$:UNTIL G$="Y" OR G$= "N" 160 IF G$="Y" THEN RUN 170 CLS:PRINT '"Bye for now.":END
RUN
Sequence generator This program produces sequences. Enter your formula involving N, for example (N*N-N+4/2 Formula involving N: ?(N*N-N +10)/2 Here are the first ten terms: 5, 6, 8, 11, 15, 20, 26, 33, 41, 50 Another go? Y or N
Here are some sequences produced by this program for various different formulae. Can you see what formula was used in each case? Check your guess by inserting the formula in the above program. (The answers are given later on in this section and further examples are given in the subsequent sections.)
(a) 1, 6, 11, 16, 21, 26, 31, 36, 41, 46
(b) 1, 2, 4, 8, 16, 32, 64, 128, 256, 512
(c) 1, 4, 9, 16, 25, 36, 49, 64, 81, 100
(d) 1, 2, 4, 7, 11, 16, 22, 29, 37,46
(e) 4, 4, 8, 12, 20, 32, 52, 84, 136, 220
The formulae for the first few sequences are not too difficult to determine. For (a) it is 5*N - 4, for (b) it is 2N/2 while for (c) it is N*N. The fourth one (d) is not quite so easy to guess, it is (N*N - N + 2)/2. Finally, the formula for the fifth one (e) is impossible to guess unless you've seen it before, in fact it is:
4*INT((0.5 + SQR(5)/2)N - (0.5 - SQR(5)/2)N)/SQR(5))
Here are some other formulae that you might like to try out.
1 + (-1)N
N * (-1)N
INT( SIN(N) * 10)
A + (N - 1)*D
where A is the first term of the sequence and D is the common difference. Here are some further examples of arithmetic sequences.
5, 10, 15, 20, 25, 30, 35, 40, 45, 50 | (A = 5, D = 5) | |
1, 1.5, 2, 2.5, 3, 3.5, 4, 4.5, 5, 5.5 | (A = 1, D = 0.5) | |
0, 2, 4, 6, 8, 10, 12, 14, 16, 18 | (A = 0, D = 2) |
Listing 5.2
LIST
10 REM Arithmetic sequences 20 MODE 1:COLOUR 3:PRINT ' TAB(10);"A rithmetic sequences"':@%=10 30 PRINT "This produces arithmetic se quences."' 40 PRINT "Enter the first term, the c ommon differ-ence and the number of term s required."' 50 COLOUR 1:INPUT "First term: ";A 60 INPUT '"Common difference: ";D 70 REPEAT 80 INPUT '"Number of terms: ";N 90 IF N<1 OR N<>INT(N) THEN COLOUR 3 :PRINT '"Try again please.":COLOUR 1 100 UNTIL N>0 AND N=INT(N) 110 COLOUR 1:PRINT '"Here is the seque nce:"':COLOUR 2 120 TERM=A:SUM=0 130 FOR I=1 TO N 140 IF 38-POS<LEN(STR$(TERM)) THEN PR INT 150 PRINT ;TERM;:IF I<N THEN PRINT ", "; 160 SUM=SUM+TERM:TERM=TERM+D 170 NEXT 180 PRINT ''"The sum is: ";SUM 190 COLOUR 3:PRINT CHR$(7) ' TAB(10);" Another go? Y or N "; 200 REPEAT:G$=GET$:UNTIL G$="Y" OR G$= "N" 210 IF G$="Y" THEN RUN 220 CLS:PRINT '"Bye for now.":END
RUN
Arithmetic sequences This produces arithmetic sequences. Enter the first term, the common differ- ence and the number of terms required. First term: ?5 Common difference: ?5 Number of terms: ?10 Here is the sequence: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50 The sum is: 275 Another go? Y or N
First offer | Second offer | |
First year | (First 6 months £3000) | |
(Second 6 months £3120) | ||
£6120 | £6120 | |
Second year | (First 6 months £3240) | |
(Second 6 months £3360) | ||
£6600 | £6360 | |
Third year | (First 6 months £3480) | |
(Second 6 months £3600) | ||
£7080 | £6600 |
You can see that if you were to stay at the job for more than one year then the first offer is better. Notice that in the first offer the amount received during the first 6 months each year increases by £240, and so the annual increase is in fact £480. The annual salary for the first offer fits into the arithmetic sequence with the formula
6120 + (N - 1)*480
while the second has the formula
6120 + (N - 1)*240
Suppose you were given a third alternative: 'Start with £1440 a quarter with a rise of £60 at the end of every 3 months.' Which offer would you prefer now? Hopefully the answer should be clear to you. The calculation would go as follows:
Third offer | ||
First year | (First 3 months £1440) | |
(Second 3 months £1500) | ||
(Third 3 months £1560) | ||
(Fourth 3 months £1620) | £6120 | |
Second year | (First 3 months £1680) | |
(Second 3 months £1740) | ||
(Third 3 months £1800) | ||
(Fourth 3 months £1860) | £7080 |
2, 6, 18, 54, 162
where every term (except the first) is three times the previous term. The general formula for a geometric sequence is given by
A * R(N-1)
where A is the first term and R is the common ratio.
Here are some further examples of geometric sequences.
4, 2, 1, 0.5, 0.25, 0.125, 0.0625 (A = 4, R = 0.5)
2, -4, 8, -16, 32, -64, 128 (A = 2, R= -2)
The next program may help you analyse geometric sequences. You enter the first term of the sequence, the common ratio and the number of terms required. Notice that a formula is not required since the computer does the calculation iteratively. In addition the program adds up all the terms in the sequence and gives you the answer.
Listing 5.3
LIST
10 REM Geometric sequences 20 MODE 1:COLOUR 3:PRINT ' TAB(10);"G eometric sequences"':@%=10 30 PRINT "This program creates geomet ric sequences " 40 PRINT "Enter the first term, the c ommon ratio and the number of terms requ ired."' 50 COLOUR 1:INPUT "First term: ";A 60 INPUT '"Common ratio: ";R 70 REPEAT 80 INPUT '"Number of terms: ";N 90 IF N<1 OR N<>INT(N) THEN COLOUR 3 :PRINT '"Try again please.":COLOUR 1 100 UNTIL N>0 AND N=INT(N) 110 COLOUR 1:PRINT '"Here is the seque nce:"':COLOUR 2 120 TERM=A:SUM=0 130 FOR I=1 TO N 140 IF 38-POS<LEN(STR$(TERM)) THEN PR INT 150 PRINT ;TERM;:IF I<N THEN PRINT ", "; 160 SUM=SUM+TERM:TERM=TERM*R 170 NEXT 180 PRINT ''"The sum is: ";SUM 190 COLOUR 3:PRINT CHR$(7) ' TAB(10);" Another go? Y or N "; 200 REPEAT:G$=GET$:UNTIL G$="Y" OR G$= "N" 210 IF G$="Y" THEN RUN 220 CLS:PRINT '"Bye for now.":END
RUN
Geometric sequences This program creates geometric sequences Enter the first term, the common ratio a nd the number of terms required. First term: ?2 Common ratio: ?-2 Number of terms: ?8 Here is the sequence: 2, -4, 8, -16, 32, -64, 128, -256 The sum is: -170 Another go? Y or N
100 + 100 * 6/100
= 100 + 100*0.06
= 100 * 1.06
which is £106. At the end of two years she will have
100 * 1.06 + (100 * 1.06) * 0.06
= 100 * 1.06 * 1.06
and you should be able to observe that after 10 years she will have
100 * 1.0610.
The total amount at the end of each year forms a geometric sequence as shown below.
100 * 1.06, 100 * 1.062, 100 * 1.063, 100 * 1.064
100 * 1.0655, 100 * 1.0656, 100 * 1.0657, 100 * 1.0658
100 * 1.0659, 100 * 1.06510
More generally, if you start with an amount A and receive interest 1% per annum then after N years your original amount has grown to the following amount:
A * (1 + 1/100)N
1000 * 1.06
assuming that no further deposits or withdrawals are made. If, on the other hand, the bank paid interest every 6 months (and paid interest on the interest given, ie compounded the interest) then the total at the end of the year would be
1000 * (1.03)2.
More generally, if the bank paid 6% interest compounded N times a year then at the end of one year the £1000 would grow to:
1000 * (1 + 0.06/N)N
The table below illustrates the different amounts depending on how often interest is compounded.
N | 6% compounded | Total at end of year | |
(to nearest penny) | |||
1 | yearly | £1060.00 | |
2 | semiannually | £1060.90 | |
4 | quarterly | £1061.36 | |
6 | bimonthly | £1061.52 | |
12 | monthly | £1061.68 | |
52 | weekly | £1061.80 | |
365 | daily | £1061.83 | |
8760 | hourly | £1061.84 |
Listing 5.4
LIST
10 REM Compound interest via formula 20 MODE 1:COLOUR 3:PRINT ' TAB(11);"C ompound interest"':@%=&02020A 30 PRINT "This program shows the effe ct on `1000 at 6% compounded a number o f times per year."'' 40 PRINT "Enter number of times inter est is to be compounded.":COLOUR 1 50 REPEAT 60 INPUT '"Number: ";N 70 IF N<1 OR N<>INT(N) THEN COLOUR 3 :PRINT '"Try again please.":COLOUR 1 80 UNTIL N>0 AND N=INT(N) 90 T=1000*(1 + 0.06/N)^N 100 COLOUR 1:PRINT '"The compounded va lue is ";:COLOUR 2:PRINT ;T 110 COLOUR 3:PRINT CHR$(7) '' TAB(10); "Another go? Y or N "; 120 REPEAT:G$=GET$:UNTIL G$="Y" OR G$= "N" 130 IF G$="Y" THEN RUN 140 CLS:PRINT '"Bye for now.":END
RUN
Compound interest This program shows the effect on `1000 at 6% compounded a number of times per year. Enter number of times interest is to be compounded. Number: ?52 The compounded value is 1061.80 Another go? Y or N
You could make some additions and alterations to allow for other interest rates. Such a program appears next.
Listing 5.5
LIST
10REM Compound interest via formula 20MODE 1:COLOUR 3:PRINT ' TAB(11);"Co mpound interest"':@%=&02020A 30PRINT "This program shows the effec t on `1000 at 6% compounded a number of times per year."'' 40PRINT "Enter the interest rate.":CO LOUR 1 50REPEAT 60INPUT '"Interest rate: ";I 70IF I<=0 OR I>=100 THEN COLOUR 3:PRI NT '"Be reasonable.":COLOUR 1 80UNTIL I>0 AND I<100 90COLOUR 3:PRINT '"Enter number of ti mes interest is to be compounded.":COLOU R 1 100REPEAT 110 INPUT '"Number: ";N 120 IF N<1 OR N<>INT(N) THEN COLOUR 3: PRINT '"Try again please.":COLOUR 1 130UNTIL N>0 AND N=INT(N) 140T=1000*(1 + I/100/N)^N 150COLOUR 1:PRINT '"The compounded val ue is ";:COLOUR 2:PRINT ;T 160COLOUR 3:PRINT CHR$(7) '' TAB(10);" Another go? Y or N "; 170REPEAT:G$=GET$:UNTIL G$="Y" OR G$=" N" 180IF G$="Y" THEN RUN 190CLS:PRINT '"Bye for now.":END
RUN
Compound interest This program shows the effect on `1000 at 6% compounded a number of times per year. Enter the interest rate. Interest rate: ?10 Enter number of times interest is to be compounded. Number: ?52 The compounded value is 1105.06 Another go? Y or N
Stake | Loss | Win | |
First toss | £1 | £1 | |
Second toss | £2 | £2 | |
Third toss | £4 | £4 | |
Fourth toss | £8 | £8 | |
Fifth toss | £16 | £16 | |
TOTAL | £15 | £16 | |
NET GAIN = £1 |
(e) 4, 4, 8, 12, 20, 32, 52, 84, 136, 220
which is given by the formula
4*INT(((0.5 + SQR(5)/2)N - (0.5 - SQR(5)/2)N) / SQR(5)).
Rather than use this formula there is a more obvious way of creating the sequence. Every term except the first two is the sum of the two previous ones.
4 + 4 = 8
4 + 8 = 12
8 + 12 = 20
and so on.
Sequences of numbers created in this way we called Fibonacci sequences. It was in 1202 that Leonard of Pisa, nicknamed Fibonacci, observed such a sequence of numbers associated with the breeding of rabbits.
Here are two other Fibonacci sequences:
2, 5, 7, 12, 19, 31, 50
3, 5, 8, 13, 21, 34, 55
The next program will produce Fibonacci sequences ad nauseum.
Listing 5.6
LIST
10REM Fibonacci sequences 20MODE 1:COLOUR 3:PRINT ' TAB(10);"Fi bonacci sequences"':@%=10 30PRINT "This program creates Fibonac ci sequences "' 40PRINT "Enter two integers, separate d by a comma":COLOUR 1 50REPEAT 60 INPUT '"Two numbers: ";U,V 70 IF U<>INT(U) OR V<>INT(V) THEN COL OUR 3:PRINT '"Integers please.":COLOUR 1 80UNTIL U=INT(U) AND V=INT(V) 90REPEAT 100 INPUT '"How many terms do you want : ";N 110 IF N<1 OR N<>INT(N) THEN COLOUR 3: PRINT '"Try again please.":COLOUR 1 120UNTIL N>0 AND N=INT(N) 130COLOUR 1:PRINT '"Here is the sequen ce:"':COLOUR 2 140FOR I=1 TO N 150 IF 38-POS<LEN(STR$(U)) THEN PRINT 160 PRINT ;U;:IF I<N THEN PRINT ", "; 170 W=U+V:U=V:V=W 180NEXT 190COLOUR 3:PRINT CHR$(7) '' TAB(10);" Another go? Y or N "; 200REPEAT:G$=GET$:UNTIL G$="Y" OR G$=" N" 210IF G$="Y" THEN RUN 220CLS:PRINT '"Bye for now.":END
RUN
Fibonacci sequences This program creates Fibonacci sequences Enter two integers, separated by a comma Two numbers: ?3,5 How many terms do you want: ?7 Here is the sequence: 3, 5, 8, 13, 21, 34, 55 Another go? Y or N
Here is an exercise that you may like to do. Write a short program for your computer.
'Write down any two integers. Form the Fibonacci sequence by adding pairs of terms to form a third term. Find the ratio of each term in the sequence with the one immediately before it. What happens to this ratio as the number of terms gets large? Work out the value of 0.5 + SQR(5)/2'.