Quandaries and Queries
Problem: Find the first 4 terms and the nth term of the infinite sequence defined recursively as follows:
Note: Quantities in brackets are subscripts
Using the recursive formula, the first 4 terms are;
The nth term a(n) = 2n-1 x 3 (equation 1)
Equation 1 must be proven using mathematical induction. This is where I am having a problem.
Thus, assume P(n) is true. a(n) =2n-1 x 3 ( induction hypothesis)
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Would appreciate your assistance. Thanks
The recursive definition says that the first term is 3 and after that every term is twice the preceding term. You think you have an expression for a with any subscript. The expression is
You want to prove this by induction so you first show the expression is valid when the subscript is 1. That you did. For the second step you assume that the expression is valid for some particular value of the subscript and you want to show it is valid for the next value of the subscript. Thus at this point you know:
So what is an+1?
Hence, by induction, the expression is valid for all subscripts which are integers greater than or equal to one.