Sequences can be defined recursively: one or more terms are...

Sequences can be defined recursively: one or more terms are given explicitly; the remaining ones are then defined in terms of their predecessors.Give the first six terms of the sequence and then give the $n$ th term.
$$a_{1}=1 ; \quad a_{n+1}=a_{n}+\cdots+a_{1}$$

Key Concepts

Recursively Defined Sequences

A recursively defined sequence is one in which certain starting terms are given explicitly and each subsequent term is defined using previous terms in the sequence. This type of definition often involves establishing a recurrence relation that dictates how the sequence evolves over time, providing a basis for both iterative computation and deeper analysis.

Recurrence Relations

A recurrence relation is an equation that expresses each term of a sequence as a function of its preceding terms. Understanding how to set up and analyze recurrence relations is crucial for solving problems involving sequences, as it allows for the determination of subsequent elements and can lead to the derivation of closed-form expressions.

Closed-form Expressions

Deriving a closed-form expression for a sequence involves finding an explicit formula that directly calculates any term in the sequence without the need for recursion. This can simplify the analysis of the sequence, particularly for determining long-term behavior or computing terms that would otherwise require extensive recursive calculation.

Geometric Growth

In the context of sequences defined by cumulative sums of previous terms, recognizing patterns such as geometric growth can be pivotal. When the sequence doubles at each step, it indicates exponential behavior, which is characteristic of geometric sequences and facilitates the derivation of the nth term through the identification of common ratios.

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