The Method of Difference to Sum a Sequence

Sequences and Series is an integral chapter in algebra. In this chapter, you will learn how to work with progressions. In other words, you will learn how to arrange given values into a sequential order based on specific characteristics shared by the terms. 

In particular, you will focus on Arithmetic Progression, Arithmetic Mean, Geometric Progression, Geometric Mean, Harmonic Progression, and Arithmo-Geometric Mean. It is one of the conceptual chapters in various competitive exams.

Sequence

Sequences in math are a range of numbers that follows a certain well-ordered and well-defined pattern. In such a sequence, the order in which the terms appear matters. 

You can determine the sum of the series using partial fractions’. And consequently, define the common ratio shared between the terms and the number of terms in the given series this way as well. It is also called the method of difference to sum a sequence. 

For instance, consider the following example:

1, 2, 3, 4, 5, 6, 7, 8, 9

These terms are said to be an arithmetic progression as they share a common difference between them, which in the above case is equal to 1. Consider the following sequence:

0, 5, 10, 15, 20, 25, 30

This is also a sequence in arithmetic progression. The common difference between the terms is equal to 5. However, the following:

2, 4, 8, 16, 32, 64

These terms are said to be in Geometric Progression. The common ratio here is equal to 2. 

Method of Difference to Sum a Sequence

We can use a few methods to find the sum of a given sequence when it is in progression. The method of difference to sum a sequence is one such method. It is one of the simplest techniques to arrive at the required sum without fully calculating the sum for the series. This technique is used to cancel out terms using the concept of partial fractions in algebra. 

This technique can be used only if the general term of the series is in the form of:

f(k+1)-f(k)

Where, f() denotes a function. 

Example:

Find the value of: ∑r=nr=31(r+1)(r+2)

Solution:

Using Partial Fractions method we get:

1(r+1)(r+2)= Ar+1+Br+2Further simplifying:

Or, 1 ≡ A(r+2)+ B(r+1)

Substituting values of A and B, we get:

Assuming r = – 1

We get,

1= A(- 1 + 2 ) 

A=1

Assuming r = – 2

We get,

1= B (- 2 + 1 ) 

B= – 1

Now by applying the method of difference to sum a sequence, we get:

1 / (r + 1 ) ( r + 2 ) = 1/ ( r + 1 ) – 1 / ( r + 2 )

By replacing the values, we’ll find the terms canceling out to give the result:

 ∑r=nr=3( 1 / ( r + 1 ) ( r + 2 )) = ¼ – (1/(n+2))

Or, (n-2)/4*(n+2)- This is the required sum of the geometric progression given.

Geometric Series Formula

The geometric series formula can find the nth positional term in the sequence. In this regard, we will require the first term and the common ratio shared between the terms. 

If no common ratio is provided, we can calculate it by taking the ratio of any of the two consequent terms. 

The Geometric Series Formula is given by:

an = arn-1

where, 

• ‘a’ is the first term
• ‘r’ is the common ratio
• ‘n’ is the positional value of the term that we want to find 

Sum of Geometric Series

The Sum of Geometric Series refers to the sum of all the digits in a given geometric progression series. We must first determine whether it is a finite or infinite geometric series to calculate this value. In both these cases, different formulae are used:

• Finite Geometric Series: 

If the number of terms in a given series can be mathematically determined, it is a finite series. In this case, you can use the following formula to calculate its sum: 

Sn = a(1−rn)/(1−r) given that r≠1; and, 

Sn = an when r = 1.

Where 

• ‘Sn’ is the sum of the terms in geometric progression up until the nth term
• ‘a’ represents the first term
• ‘r’ is the common ratio  
• ‘n’ is the positional value of the term which we want to find 

• Infinite Geometric Series:

A geometric series is an infinite GP series if the total number of terms in the given series can’t be mathematically calculated. In such cases, you can calculate the sum using the following formula:

Given that |r| < 1 

S∞ = a/(1 – r)

• Where’S∞’ is the sum of the terms in geometric progression up until the nth term
• ‘a’ represents the first term
• ‘r’ is the common ratio  

Moreover, you should note that if the value of |r| is greater than 1, the series is divergent. In other words, there is no sum for the series as it does not converge. 

Conclusion

To conclude, we can understand that any number series with a common ratio is in geometric progression. Any given sequence that is said to be in geometric progression has a sum of terms upto infinity when the common ratio is less than one. However, no sum is possible if the series is divergent. This sum can be calculated via multiple methods, including the methods of difference, as discussed above. It is a shortcut method using partial fractions.