Using the recursive formula for each arithmetic sequence,...

Using the recursive formula for each arithmetic sequence, find the second, third, and fourth terms of the sequence. Then write the explicit formula that represents the sequence.
$A(n)=A(n-1)-2 ; A(1)=0$

Key Concepts

Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This constant difference, known as the common difference, allows the sequence to be generated by adding (or subtracting) the same value to the previous term, leading to a predictable pattern that can be expressed either recursively or explicitly.

Recursive Formula

A recursive formula defines each term of a sequence in relation to one or more previous terms. In the context of an arithmetic sequence, the recursive formula typically specifies that a given term is the previous term plus or minus the common difference. This approach requires an initial term to provide a starting point for generating subsequent terms.

Initial Condition

The initial condition provides the starting value of the sequence, allowing the recursive formula to be applied. In arithmetic sequences, the initial term is crucial since it determines the specific values in the sequence when combined with the common difference. It acts as the base case from which all other terms are derived.

Explicit Formula

An explicit formula defines the nth term of a sequence directly as a function of n, without reference to previous terms. For arithmetic sequences, the explicit formula is typically expressed in terms of the initial term and the common difference, enabling direct computation of any term in the sequence without iterative calculation.

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