Friday, March 15, 2013

Finding the n th term in a sequence

We encounter situations to find the nth term of a series that is neither in arithmetic progression nor in geometric progression, and here we use the below simple technique to figure out the nth term.

Say we have a sequence in which the nth term is to be figured out. If the sequence of numbers are in Arithmetic progression or Geometric progression or Harmonic progression we have direct formula to find the     nth term. But if the series follows none of the above progressions, even then we have a simple method to calculate the nth term applying basic mathematical tools.

Say the sequence is 

10  12  16  22  30  . . . . .  

We shall consider the general expression aN+ bN + c 

Where N denotes the term number and a,b,c are constants, now we can generate equations using the general expression as shown below:



Equation 1 : Put N = 1 in general expression and equate it to 10 ( first term )          a+ b +c = 10
Equation 2 : Put N = 2 in general expression and equate it to 12 ( second term )   4a+2b+c = 12
Equation 3 : Put N = 3 in general expression and equate it to 16 ( third term )       9a+3b+c = 16

Now solving the above three equations we obtain the values of constants a,b and c.
Solving of equations may be done using matrices or linear equation formula. 

By elimination method Eliminating c in Equation 1 and 2 (Subtracting eq 2 from eq 1) we get  3a + b = 2
Again Eliminating c in Equation 2 and 3 (subtracting eq 2 from 3) we get 5a +b = 4

Now solving the equations thus obtained, 3a+b= 2 and 5a+b=4.

Eliminating b from below equations ...
5a + b = 4
3a + b = 2 

We get 2a = 2 and a =1 from the above equations hence finding b value from the same equation b = -1 
and similarly c = 10

The general expression becomes N-N + 10
and thats the nth term of the given series. 
The same way one can find the nth term for any series. 





Monday, February 11, 2013

Pal Multiplication. . .

Wanna multiply 99 with another number, do it simpler as shown below

Any two digit number can be multiplied with 99 in no time. Let us look this with the help of a simple example.

Say you want to multiply 54 and 99. All you need to do is simple subtraction.

We know that :   56  X  99  =  5544 ( by usual multiplication )

Step 1 : Reduce 1 from the number to be multiplied with 99 and find the answer.   56 -1 = 55.
                                   
Step 2 : Subtract the obtained answer from 99 and calculate the balance.  99- 55 = 44 

Step 3 : Place both answers of step 1 and 2 side by side in order, you end up with the product easily.
             Answer = 5544

Here the answers of step one and two are treated as friends and they sit adjacent to each other always and so this multiplication got the name 'pal multiplication'. That was cool ...  :)

Lets look at another case, say 21 is to be multiplied with 99

  1. Firstly subtract 1 from 21.  21-1 = 20.
  2. Now subtract 20 from 99.  99-20 = 79
  3. Product = 2079 ;  Also we know that 21 x 99 = 2079. 

NOTE : Please do remember that this simple technique is useful in calculating 2 digit numbers only.

Saturday, February 9, 2013

Special Numbers

Special numbers have always been identified and appreciated by mathematicians over the years. They usually exhibit peculiarity which highlight their presence in manipulation. By chance we might have seen such numbers in our day to day practice, which have not come into light. Now we shall look at numbers of the same sort.....

376 - Genetic Multiplication 

Number 396 is so special that, when we multiply and number ending with 376 ( both multiplier and multiplicand ) we obtain a product ending with 376.

For instance :                  376  x  4376    =   1645376
                                      376  x  8376    =   3149376
                                      376  x  3376    =   1267376

This is like children inheriting the genes of the parents . 


Magic Multiplication 

The special number 8547 gives amazing results on multiplication of 13 multiples as seen below 

8547  X  13  =  111111
8547  X  26  =  222222
8547  X  39  =  333333
8547  X  52  =  444444
8547  X  65  =  555555
8547  X  78  =  666666
8547  X  91  =  777777

and so on .....

Also the number 15873 gives similar results on multiplication with 7 multiples as shown  ... 

15873  X  7    =  111111
15873  X 14   =  222222
15873  X 21   =  333333
15873  X 28   =  444444
15873  X 35   =  555555


Reversing !! Its Ridiculous 

We might have heard / seen that 1089 is such a tricky number, now we see it lively here ..... 

1089 + 8019 = 9108

1089  X 1   = 1089  <=>  9801  =  9  X 1089
1089  X 2   = 2178  <=>  8712  =  9  X 1089
1089  X 3   = 3267  <=>  7623  =  9  X 1089
1089  X 4   = 4356  <=>  6534  =  9  X 1089

The symbol <=> signifies reversing the digits ... This stuff seems something like our 9 table ...

Please post your queries if any.... Thanks for your interest in Math made easy...



Tuesday, February 5, 2013

Marshal art Multiplication

:D :D There is a special number 153846 for which we have got some cheeky manipulations.

If we multiply the number by 4, the product will be 615384


If this multiplier 4 kicked the 6 from units place to the lakh's place... and so we call is Marshal Art multiplication ....  Things of this kind seem funny in math, there is fun in science too  :)

Sunday, February 3, 2013

Half and Laugh Multiplication

Now multiply any number with 5 in no time.....
Usually we take a 2 digit number and multiply by 5, end up with a solution in some time. 

In another logical sense also we can find equal product saving much time. 

For odd numbers : 
Make half of the multiplicand and remove the point from the answer obtained

say the number is 483, we know that half of 241.5 and now removing the point we obtain 2415 
also 483 x 5 = 2415

For Even numbers :
Make half of the multiplicand and put zero after the end of the answer obtained. 

say the number is 626, half of it is 313 and putting a zero makes it 3130, which gives the product in a few seconds.

That was cool, I guess ........ :)