Saturday, September 27, 2014

Use mathematical induction to find a formula for the sum of the first n terms of the sequence.


Now let's denote the first term of the above sequence by and k'th term by and let's write down the first few sums of the sequence.






From the above sequence it appears that the formula for the sum of the k terms is,



Now the difference between the consecutive terms of the series is 4, so we can write down ,






Now...


Now let's denote the first term of the above sequence by and k'th term by and let's write down the first few sums of the sequence.






From the above sequence it appears that the formula for the sum of the k terms is,



Now the difference between the consecutive terms of the series is 4, so we can write down ,






Now we can verify that it is valid for n=1,


Plug in k=1 to verify,



Let's assume that the formula is valid for n=k and now we have to show that it is valid for n=k+1











So the formula is valid for n=k+1 also, 


So the formula can be written as ,


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