Let the sequence $a_{n}$ be defined as follows: $a_{1}=1, a_{n}=a_{n-1}+2 \text { for } n \geq 2 .$ Find first five terms and write corresponding series.

Answer: The first five terms are $1, 3, 5, 7, 9$ . The corresponding series is $1 + 3 + 5 + 7 + 9$ .

Step-by-step solution

Step 1: Understand the Sequence Definition

The problem defines a sequence where the first term, $a_1$ , is 1. For any subsequent term $a_n$ (where $n$ is greater than or equal to 2), it is found by adding 2 to the previous term, $a_{n-1}$ . This means it's an arithmetic progression.

Step 2: Calculate the First Five Terms

We start with $a_1 = 1$ . To find $a_2$ , we add 2 to $a_1$ . We continue this process, adding 2 to the previous term, to find $a_3$ , $a_4$ , and $a_5$ .

Step 3: Write the Corresponding Series

A series is the sum of the terms of a sequence. To write the series corresponding to the first five terms, we simply add them together.

Step 4: Calculate the Sum of the Series (Optional)

Although not explicitly asked, we can calculate the sum of the first five terms of the series by adding them up. This sum is 25.

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Let the sequence ana_{n}an​ be defined as follows: a1=1,an=an−1+2 for n≥2.a_{1}=1, a_{n}=a_{n-1}+2 \text { for } n \geq 2 .a1​=1,an​=an−1​+2 for n≥2. Find first five terms and write corresponding series.