How does an existing observable pattern in a sequence of numbers help you extend that sequence?
2007-02-09 01:59:24 · 4 answers · asked by Anonymous in Science & Mathematics ➔ Mathematics
Just by inspection you can guess what is the pattern of a sequence. it can be then extrapolated in the same manner. This helps in 90% of the problems. Take for example the Arithmetic progression , such as 3, 5, 7, 9, 11, 13 ...... and so on. It is easily observed that the terms are increasing by 2 above the previous term. So we could rewrite, the sequence 3, 3+2, 3 + 2x2, 3 + 3x2, 3 + 4x2 ........ and so on. More formally this can be written as Tn = 3 + (n - 1 )x2 if we observe that there are n - 1 2's in n-th place. Thus we could find out 13 th term , just put 13 in place of n.
Take another example, geometric series, 5, 10 , 20, 40, 80........ We can see that each term is just double of the previous; so that we can put Tn = 5x2^(n - 1) for n-th term and calculate any term just by putting the value of n in the formula.
Now take another example, 1, -1, 1, -1, 1, -1,......... What do we observe ? That Tn = (-1)^(n - 1) . This is called an alternating sequence.
Why this n-1 ?. It is because the 'pattern' of the sequence is same whatever the 1st term might be. The characteristics are evident only from the second member onwards.
Still another example, 1, 1, 2, 3, 5, 8, 13, 21, ......... Observe that any term is the sum of the previous two, barring the first; of course. This is a Fibonacci sequence Beautiful and interesting. It encodes a golden ratio, Nature's most favoured ratio.
So far it is about simple sequences. in one more slightly complicated case, we can form complex sequences, e.g. having compound character of two kinds of sequences in one, e.g. arithmetico-geometric series. Still in other patterns, we may fix some , say, first few terms as we wish , without any evident pattern, and thenafter give a regular patter to the sequence.
Go to books, go to the web, ponder more and have nice revealations.
2007-02-09 02:26:40 · answer #1 · answered by Anonymous · 0⤊ 0⤋
Its depend upon your logical power.
2007-02-09 10:05:52 · answer #2 · answered by yogesh gulhania 2 · 0⤊ 0⤋
http://etext.lib.virginia.edu/cgi-local/DHI/dhiana.cgi?id=dv3-50
go here it might explain it.
2007-02-09 10:08:58 · answer #3 · answered by MissPanda 1 · 0⤊ 0⤋
I need an example to give you an answer. Thanks
2007-02-09 10:02:22 · answer #4 · answered by yummymummy 3 · 0⤊ 0⤋