Chapter 8: Problem 259

What is meant by forward and backward shifting?

Short Answer

Expert verified

Forward and backward shifting are techniques used in solving problems involving sequences or series by creating new sequences through the substitution of terms. In forward shifting, each term is replaced with the successor term, while in backward shifting, each term is replaced with the predecessor term. These methods help identify patterns and simplify calculations in problems involving sequences and series.

Step by step solution

Definition of Forward Shifting

Forward shifting involves creating a new sequence by replacing each term with its successor term. That is, the (n+1)th term is substituted for the nth term in the original sequence. This technique helps expose patterns or make calculations simpler.

Definition of Backward Shifting

Backward shifting is the process of creating a new sequence by replacing each term with its predecessor term. That is, the (n-1)th term is substituted for the nth term in the original sequence. Backward shifting can also help identify patterns and simplify calculations.

Example of Forward Shifting

Suppose we have a sequence \(a_n\) which is defined by the formula: \(a_n = 3n - 2\). Forward shifting the sequence entails replacing each term with its successor or the (n+1)th term: \(a_{n+1} = 3(n+1) - 2 = 3n + 1\) Now, the new sequence \(a_{n+1}\) is: \(a_1 = 1, a_2 = 4, a_3 = 7, a_4 = 10, \dots\) This forward shifted sequence reveals a simple arithmetic pattern.

Example of Backward Shifting

Let's use the same sequence \(a_n = 3n - 2\). To backward shift the sequence, we replace each term with its predecessor or the (n-1)th term: \(a_{n-1} = 3(n-1) - 2 = 3n - 5\) Now, the new sequence \(a_{n-1}\) is: \(a_0 = -2, a_1 = 1, a_2 = 4, a_3 = 7, \dots\) The backward shifted sequence also shows an arithmetic pattern, albeit with a different starting point. In conclusion, forward and backward shifting help in revealing patterns and simplifying calculations in problems involving sequences and series. The choice of which shifting method to use depends on the problem and the goal of the analysis.

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