Calculating the squares of large numbers can be a difficult
task but we have found some relations between the numbers that can easily
reduce the calculation time and help evaluating the squares

n

n

^{2 }= (n+1) (n-1) +1i.e Square of a number= (Preceding number)(Succeeding Number)+1

For e.g.:- n=99

99

^{2}= (99-1) (99+1) + 1
= 98*100 + 1

= 9801

Hence, now we know that square of 99 = 9801

The other relation is:

n

^{2}= (n-1)

^{2 }+ (n-1) + n

i.e square of a number= Square of preceding number+Preceding Number+The number itself

For e.g.:- n=13

13

^{2 }= (13-1)^{2 }+ (13-1) +13
= 144 +25

= 169

We can combine the above two identities to form another useful relation

Suppose, we have n=98

We know that 99

^{2}= 9801 [i.e. (n+1)^{2 }=9801] from the identity no.1. So, by using the identity no.1 and no.2 we form a relating:-(n+1)

^{2 }= n

^{2 }+n + (n+1)

Hence, we can conclude that,

n

^{2 }= (n+1)^{2}- n – (n+1)By putting n=98

= (99)

^{2}- 98 - 99
= 9801 - 98
- 99

= 9604

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