The Method of Differences

The method of differences is a trick method to solve those sequence and series problems that do not come in the categories of traditional progression series. We can easily solve an AP and a GP series by applying the respective formulas for the summation to find the value of the nth term of the series, but for a series like 1, 5, 11, 27,.. (not an AP or GP series), solutions cannot be found from using direct formulas. To solve such a series, we use the method of differences.

Method of Differences

While calculating sequence and series problems, we often come across such series that don’t fall in the category of arithmetic progression or geometric progression. These series do not follow a constant or regular pattern of progression and, therefore, can not be calculated by using traditional methods of using a formula-based approach.

The method of differences was developed to solve such a series. Under this method of solving sequence and series problems, we try to get an AP or a GP series by subtracting the subsequent terms of the series.

The following is the step-by-step procedure to calculate a series by using the method of differences:

Step 1: First, we subtract the value of the subsequent terms of the series with the immediately preceding term; here, we do not subtract the first and the last term of the series.

Step 2: After subtracting, we get new values from the second term of the series.

Step 3: In this step, we check if the new terms form an AP or a GP series.

Step 4: When we successfully determine the nature of the series, we calculate the sum of the series for (n – 1) terms.

Step 5: Now, we have a simplified solution of the series. However, the series can be simplified by expanding it further.

Solved Examples

Let us understand how the method of differences works by looking at the following examples:

Example 1: Find the sum of n terms of the following series:

7, 9, 13, 19, 27,…., an

Solution: Since we can see that the given series is neither an AP nor a GP series, we will use the method of differences to find the sum of this series.

The given series, 

S = 7, 9, 13, 19, 27,…, an

To use the method of differences, we subtract the subsequent terms of the series,

S = 7 + 9 + 13 + 19 + 27 +,…. + an

             S =       7 + 9 + 13 + 19 + 27 +,…. + an – an-1 + an

We get, 0 = 7 + 2 + 4 + 6 + 8 + …. + an – an-1 – an

Or, an = 7 + { 2 + 4 + 6 + 8 + …. + an – an-1}

The function in RHS makes an AP with n-1 terms, and the first term 2 and d = 2.

Using the summation formula for AP,

an = 7 + [(n-1)2{ 2.2 + (n-1-1).2}

an = 7 + [(n-1)2 [ 4 + 2n – 4]

an = 7 + (n2 – n)

Further expanding this solution, we get,

an = 7n + [ n.(2n + 1)(n + 1)6] – n.(n+1)2

After calculating we get,

an =n6.(2n2 + 40)

The above expression gives the solution for the series.

Example 2: Consider the following series of numbers:

5, 17, 37, 65,…

Solution: As we can see, the series itself is neither AP nor GP.

Therefore we will be using the method of difference to calculate the sum of n numbers of the series.

First subtracting the original equation with its subsequent terms,

S = 5 + 17 + 37 + 65 + … + an

S =         5 + 17 + 37 + … + an – an-1 – an

Upon subtracting we get,

0 = 5 + 12 + 20 + 28 + … + an – an-1 – an

Or an = 5 + {12 + 20 + 28 + …+ an – an-1}

We get an AP series with n-1 terms and the first term 12 and d = 8.

Using the AP formula for summation,

an = 5 + [(n-1)/2.{2.12 + (n-1-1).8}]

an = 5 + 4n2 – 4

an = 4n2 + 1

Further expanding this expression we get,

an = 4[ n(2n + 1).(n + 1)/6] + n

an = n/3.(4n2 + 3n +14)

The above expression gives the solution for the equation.

Conclusion

The method of differences is used to solve those progression series that are not AP or GP. In this method, we subtract the terms of the series by their immediately preceding term. Subtracting the series in such a way gives a variation of an AP or GP series. The equation is further simplified by using simple algebraic functions. After expanding the values, the sum of the series can be determined using the differences method.