Let a1, a2, a3, …be a sequence defined by a1=1, an=an-1+4 ; n ⩾2 Show that an=4 n-3 for all posi

step 1 In this problem, we want to prove this statement that A .N can be found using the recursive stat

Let a1, a2, a3, …be a sequence defined by a1=1, an=an-1+4 ;...

Key Concepts

Arithmetic Sequences

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This property allows the sequence to have a predictable pattern, and it enables the derivation of an explicit formula that directly calculates the nth term without the need for recursion.

Recurrence Relations

A recurrence relation defines a sequence where each term is expressed in relation to its preceding term. In the given problem, the relation specifies that each term is obtained by adding a fixed number to the previous term. Analyzing recurrence relations helps in finding closed-form formulas and understanding the behavior of sequences over time.

Closed-form Expressions

A closed-form expression provides a direct formula to compute the nth term of a sequence without iterative or recursive steps. Converting a recurrence relation into a closed-form helps in efficiently calculating any term in the sequence and in verifying the properties of the sequence across its entire domain.

Mathematical Induction

Mathematical induction is a proof technique used to establish the validity of a statement for all positive integers. It involves verifying a base case and then proving that if the statement holds for one integer, it necessarily holds for the next integer. This method is essential for proving formulas derived for sequences defined by recurrence relations.

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