Method 3

Squares of numbers ending with 5

This method of finding the square of a number ending with five is based on the Sutra of " Ekadhikena Purvena " of respectable Vedas . It means "one more than the one before it " or " one more than the previous one .

Explaining the method by an example : -

Example 1 : - Find the square of 95 .

Answer : -

Step 1 : -

In this we simply make the square by simply multipling the digit(s) previous to 5 by one more than themselves .
i.e.
9 * (9 + 1) = 9 * 10 = 90

Step 2 : -

Now we place 25 next to the digit obtained above .
i.e.
9025
which gives us the answer.

=>Answer is 9025.

Example 2 : - Find the square of 55.

Answer : -

Proceeding as above :

Step 1 : -

5 * (5 + 1) = 5 * 6 = 30

Step 2 : -

Place 25 next to it .

Answer is 3025.

Note : - We can extend the rule to any number of digits .

Example 3 : - Find the square of 125 .

Answer : -

Step 1 : -
12 * ( 12 + 1) = 12 * 13 = 156

Step 2 : -

Place 25 next to it .

Answer is 15625 .

Algebraic proof : -

Consider the expantion

(a.x + b )² = a².x² + 2a.x.b + b²

Substituting x = 10 and b = 5

(10.a + 5)² = 10².a² + 2.10.a.5 + 5²
= 100.a² + 100.a + 25
= (a² + a)100 + 25
= a.(a + 1)10² + 25

In the above a.(a + 1).100 gives a.(a + 1)/00 and + 25 gives

a.(a + 1)/25 .

which is our answer

Exercises : -

Find the square of following numbers.

1.) 35
2.) 75
3.) 215
4.) 85
5.) 165