The first term of a sequence is x1=1 . Each s\ucceeding term...

The first term of a sequence is $x_{1}=1 .$ Each succeeding term is
the sum of all those that come before it:
$$
x_{n+1}=x_{1}+x_{2}+\cdots+x_{n}
$$
Write out enough early terms of the sequence to deduce a general
formula for $x_{n}$ that holds for $n \geq 2$

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Key Concepts

Summation

Summation involves adding a series of numbers together. In the context of the sequence, the recurrence relation involves a summation of all previous terms to produce the subsequent term, making it important to understand how cumulative sums behave.

Geometric Progression

A geometric progression is a sequence where each term is obtained by multiplying the previous term by a constant factor. When the summed sequence reveals a consistent multiplicative pattern, it can often be reinterpreted or simplified as a geometric sequence, aiding in the derivation of a general formula.

Sequence

A sequence is an ordered list of numbers defined by a specific rule that determines the progression from one term to the next. In problems like this, understanding the way each term is generated is key to both recognizing patterns and developing a general formula.

Recurrence Relation

A recurrence relation defines each term of a sequence in terms of preceding terms. This type of relation is critical when the sequence’s construction involves previous elements, such as when a term is expressed as a sum of all earlier terms.

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