Let an=(1)/(2 n-1) for n=1,2,3, …Write out the first three terms of the following sequences.

step 1 Okay, so for part A, we have A sub n is equal to 1 over 2N minus 1. So for n is equal to 1, 2, a

Let an=(1)/(2 n-1) for n=1,2,3, …Write out the first three...

Let $a_{n}=\frac{1}{2 n-1}$ for $n=1,2,3, \ldots$ Write out the first three terms of the following
sequences.
$$
\begin{array}{ll}{\text { (a) } b_{n}=a_{n+1}} & {\text { (b) } c_{n}=a_{n+3}} \\ {\text { (c) } d_{n}=a_{n}^{2}} & {\text { (d) } e_{n}=2 a_{n}-a_{n+1}}\end{array}
$$

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Key Concepts

Sequences

Sequences are ordered lists of numbers that follow a specific rule or pattern, often given by a formula. In this context, knowing the explicit formula for the sequence allows the computation of individual terms and provides a structured way to understand the underlying pattern or behavior of the sequence.

Index Shifting

Index shifting involves changing the index variable in a sequence to refer to different terms relative to the original sequence. By substituting the index with expressions like n+1 or n+3, one can generate related sequences that begin at different points of the original sequence, which is a common technique in sequence manipulation.

Term-wise Operations

Term-wise operations involve applying arithmetic operations directly to each term of a sequence. This includes operations such as squaring each term or performing linear combinations (like 2a_n ? a_n+1). Such operations are fundamental in transforming sequences and analyzing how these manipulations affect the sequence’s properties.

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