Thursday, March 8, 2012

Part: 2 Number Sequence (analyzing sequence, what is the next number in the series)

Analyzing Sequence: 
Keep in mind that there is no set rule for finding the next term in a sequence of numbers. It totally depends on your ability.  So, the more you practice, the better you will be in solving these problems. Still there are some techniques which might be of help.

First, ask if the sequence looks as it is growing slowly, quickly or in between. Here is an example of a sequence that is growing quickly: 2, 9, 28, 65, 126,....
Here is an example of one the grows slowly: 2,4,6,8,10......
And here is one that's between: 2,7,10,17,26,37,50......
If the sequence is growing slowly, you can guess that it grows according to a linear formula. e.g. 2, 5, 10, 17...

If it is growing at an intermediate rate, you may try to compare it with the sequence of squares. If the sequence seems to be growing very quickly, compare it with quick- growing sequence like (n 3 )or exponents (n x ). If a straight forward comparison doesn't work, try to see the difference between each one.
2+3 5+5 10+ 7 17+9 26+ 11 37+ 13 50
We can see a pattern in the differences. They are a series of odd numbers. So, the next number in the sequence is 50 +15= 65.
Sometimes the pattern of the differences isn't helpful. Look at 3,9, 27,81......
Writing the difference gives 3+6 9+ 18 27+ 54  81
But there is another trick. Let us look at the multiples. 3 x3 9 x3 27 x3 81
So, this is a geometric progression.
Remember, these are only techniques; the more you practice, the better you will get at these sort of maths. So, let us look at another problem.

# What is the next number in the series     1, 3, 5, 10, 15, .....?
Solved: First let us list the differences: 1+(2)3+(3)6+(4)10+(5)15 
So, there is a pattern in the differences. So, the next difference should be + 6
:. Next term = 15+ 6 + 21 .Ans.

Now let us see a tougher problem.
# What is the next number in the series 1,3,11,67, .......?
Solve: Let us take the differences: 1+(2)3+(8)11+(56)67......
This doesn't seem to produce any pattern.
So, we try to find relationship between every pair of numbers.
(1,3)      3= 1x2+1
(3, 11)   11= 3x4-1
(11, 67)  67= 11x6+1
Following this pattern we may write that the next number is
67x 8-1= 536-1=535 Ans.
 It is very important to remember that given any sequence of numbers, it is possible to develop many different rules which produce the given sequence of numbers. But those many different rules may well produce different 'subsequent' numbers. So there may be various correct answers to the given problem. Usually you are expected to given the easiest answer.
Now we will introduce you to another technique. This is useful for solving many complex sequences. But remember, this will not solve all problem.

# Find the next term in the following sequence 3,9,18,30,45
Solve: It seems that the difference between each successive term goes up by three each time. The easiest way to approach such sequence is as follows:
List your numbers. Then, on the line below, list the differences of those number . On the next line, list the differences of the differences & so on till your come to a row where all the numbers are equal.
Row 1: 3      9      18      30      45      63
Row 2:     6      9        12      15      18
Row 3:          3      3          3        3
Now we work upwards. The next number in row 2 will be 15+3 = 18. So, the next number in Row 1 will be 45 +18 = 63

# Now let us try to find the next number in the following sequence 1,4,32,512,....?
Solve: Number is growing very fast. But 4/1=4, 32/4=8, So, this is not a geometric progression. We try it a bit different way. We will write rows similar to the last problem, but in each successive row, we will write the multiple between two successive terms of the previous row
Row 1: 1          4          32          512          16384
Row 2:       4          8           16             32
Row 3:             2           2             2
So, again we produce the next term in Row 2, which is 16x2= 32. Then we produce the next term in Row 1 which is 572x32= 16384 Ans.

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