Power sequence algorithm


We know how to calculate square of a number…..
Square(n)=n*n
But this square is nothing but sum of first n odd numbers.
i.e. Square(n) =Sum of first n odd nos.
=1+3+5+….+(2n-1)
This statement indicates that if we have a series of squares like 1,4,9,16,25,.. then the series formed with the consecutive differences is nothing but series of odd numbers.
————————————-
This was just simple explaination. But it is extended to any power of a number. like cube, fourt power etc.
Now we see about cubes.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 …
Do the following operations.
1. eliminate the each third number
=>1 2 4 5 7 8 10 11 13 14 16
2. Get the running sum of this series.
=>1 3 7 12 19 27 37 48 61 75 91
3. Now eliminate each second number from this series.
=>1 7 19 37 61 91
4. Find once the running sum of this series.
=> 1 8 27 64 125 216
Isn’t it the series of cubes….Similarly for any power of a number we have to find running sum that number of times eliminating numbers as shown here.
—————————————–

1. Delete each Mth number and find the series of running sum
2. Delete each (M-1)th number and find the running sum.
.
. Delete the each 2nd number and find the running sum
It is a series of power M

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