The Sūtra: ekādhikena pūrveṇa (By one more than the one before) is used for addition (x + y) and subtraction (x  y) of numbers, and also discussed for squaring (x
^{2}) of numbers, that end with 5.
The technique for addition and subtraction used by this Sūtra is similar to the conventional methods, but replaces the 'carryover' with a dot to the digit placed in the previous place to indicate its ekādhika, or +1. Due to this, the count never exceeds 10, which helps in counting fast and reduces the chances of error. However, the technique used by this Sūtra (ekādhikena pūrveṇa) should not be confused with Upasūtra 15. śūddha (
discussed here ») which uses a completely different technique.
Addition of two (or more) numbers using Sūtra 1. ekādhikena pūrveṇa:
As an illustration, let us use this Sūtra for:
28 + 165

Steps 
28 + 165 
1. 
Equate the digits of both the numbers by prefixing imaginary Zeroes in front, and start counting upwards. 
In this case,
0 2 8
+ 1 6 5


2. 
While summing the digits, place a dot on the previous digit for whichever digit caused the count greater than 10, and continue counting by removing the Ten's place. 
In this case,
0 2^{o}8
+ 1 6 5

3

3. 
While counting consider the dots in the column as an ekādhika, or +1. 
In this case,
0 2^{o}8
+ 1 6 5

1 9 3, which is the answer!

So, for a practitioner of Vedic Mathematics, for something like:
234 + 403 + 564 + 721

0 2^{o}3^{o}4
0 4 0 3
0^{o}5 6 4
+ 0 7 2 1

1 9 2 2
Thus, 234 + 403 + 564 + 721 = 1922

Again, for something like:
78924 + 27272 + 72684

0^{o}7^{o}8^{o}9 2^{o}4
0 2 7 2^{o}7 2
+ 0 7 2 6 8 4

1 7 8 8 8 0
Thus, 78924 + 27272 + 72684 = 178880

Although, it does not provide a visible marked difference, this technique comes handy when we have too many numbers to deal with, like a shopping bill. However, this technique, not only drastically reduces the chances of error, it saves time since one never has to count more than 10.
And, this Sūtra obviously works because it is only a variation of the conventional method of addition, by using the 'Dot' as a graphic mnemonic device.
Subtraction of numbers using Sūtra 1. ekādhikena pūrveṇa:
As an illustration, let us use this Sūtra for:
34  18

Steps 
34  18 
1. 
Equate the digits of both the numbers by considering imaginary Zeroes in front, and start counting upwards. 
In this case,
3 4
 1 8


2. 
While subtracting the digits, place a dot on the previous digit for wherever digit being subtracted from a smaller digit, and sum with the digit's pūrak. A 'pūrak' of a digit is its complementary, 10  x (discussed here ») 
In this case,
3 4
 1^{o}8

6
Note: The pūrak of 8 is (10  8) = 2
And, 4 + 2 = 6

3. 
Count the number of dots in each digits, and subtract further to get the answer. 
In this case,
3 4
 1^{o}8

1 6, which is the answer!

So, for a practitioner of Vedic Mathematics, for something like:
5124  3608

5 1 2 4
 3^{o}6 0^{o}8

1 5 1 6
Thus, 5124  3608 = 1516

Again, for something like:
78924  27272

7 8 9 2 4
 2 7 2^{o}7 2

5 1 6 5 2
Thus, 78924  27272 = 51652

Although this method can only subtract smaller numbers from bigger numbers, it can also be used in subtracting bigger numbers from smaller ones because x  y = (y  x), as for something like:
234  7381

7 3 8 1
 0 2 3^{o}4

7 1 4 7
Thus, 234  7381 = (7381  234) = 7147

And, this Sūtra obviously works because it is only a variation of the conventional method of subtraction, by using the 'Dot' as a graphic mnemonic device.
Squaring of numbers ending with 5 using Sūtra 1. ekādhikena pūrveṇa:
We have discussed this technique along with this Sūtra, as it is popularly referred. However, we recommend the corollary (Upasūtra 8. antyayordaśake'pi 
discussed here ») for a more generalized approach.
As an illustration, let us use this Sūtra for:
15^{2}

Steps 
15^{2} 
1. 
Remember that square of 5 is 25. This is the 2nd part of the answer. 
In this case, the 2nd part is 25  as in all cases.

2. 
Multiply the 1st part (leaving the last digit) with its ekādhika (1st part + 1). 
In this case,
The 1st part is 1, and 1 × (1+1) = 1 × 2 = 2

3. 
Join both the results to get the answer. 
In this case,
Joining 2 and 25, we get 225, which is the answer!

Let us take another example, for something like:
35^{2}

The second part is: 25
Product of 1st part, with its ekādhika: 3 × (3+1) = 3 × 4 = 12
Joining both the results, we get 1225, which is the answer!

So, for a practitioner of Vedic Mathematics, for something like:
9995^{2}

The second part is: 25
Product of 1st part, with its ekādhika: 999 × (999+1) = 999 × 1000 = 999000
Joining both the results, we get 99900025
Thus, 9995^{2} = 99,900,025

And, for something like:
10005^{2}

The second part is: 25
Product of 1st part, with its ekādhika: 1000 × (1000+1) = 1000 × 1001 = 1001000
Joining both the results, we get 100100025
Thus, 10005^{2} = 100,100,025

However, this technique may get cumbersome for bigger numbers  which is why it is discussed for 2digit and 3digit numbers. As an example, for something like:
7875^{2}

The second part is: 25
Product of 1st part, with its ekādhika: 787 × (787+1) = 787 × 788 = 620156
Joining both the results, we get 62015625
Thus, 10005^{2} = 62,015,625

But, why does it work? For this Sūtra (ekādhikena pūrveṇa), let us consider the following:

Assuming a number (x + 5)
Then, (x + 5)^{2} = (x + 5) × (x + 5)
= x^{2} + 5x + 5x + 25
= x^{2} + x(5 + 5) + 25
= x^{2} + 10x + ab
= x (x + 10) + 25
This is exactly what this Sūtra makes us do.
Note: Since the results are 'joined', (x + 10) is always adjusted for (x + 10^{0}), which is (x + 1).

This Sūtra is also discussed for division by a number ending with 9. For example, 1/19. This discussion is merely an illustration, and with its limited domain of operability, we do not advice remembering patterns (at least for such limited usability).
Lastly, please remember that, as in any other form of mathematics, the mastery of Vedic Mathematics require practice and the judgement of applying the optimal method for a given scenario  a guideline of which, is presented in
Applications »