Discovery (in May 2017) by Mathematician Vitthal B. Jadhav

VJ’s Power Lemma Let ( Lemma: - ) = ( 5 ) 𝐻 𝑈

Where

Then for

𝐻 𝑎𝑛−1 (2𝑛 − 5) (𝑘𝑛−2)𝑘=𝑎𝑘= 1

It is useful in computing nth root of perfect n

th power. Application :-

Power of Number Ending with 5

Let 5 is the number ending with 5, then

( )2 = ( 2 ( )) 2 5 25

= 2 ( ) 25 (i.e. Append 25 to 2 )

( )3 = ( 3 ( 2 5 )) 𝑎 𝑎500

4

= 3 ( 2 ) 𝑎 𝑎500

4

Where Sum of natural numbers up to 𝑎 𝑎

( )4 = ( 4 ( 3 5 𝑎 )) (2 6 ) 25

Where 𝑎 Sum of square of natural numbers up to 2 𝑎

( )5 = ( 5 ( 4 5 5 𝑎 )

4 ) 5 3

𝑎50000

16

Where 𝑎 Sum of ‘cube of natural numbers’ up to 3 𝑎