Let the sequence $a_n$ be defined as follows: $a_1 = 1$, $a_n = a_{n-1} + 2$ for $n \geq 2$. Find first five terms and write corresponding series.

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> [!TLDR]
> The sequence is arithmetic with first term $1$ and common difference $2$, giving terms $1, 3, 5, 7, 9$. The corresponding series is $1 + 3 + 5 + 7 + 9 + \cdots$.

The definition tells us two things: where to start ($a_1 = 1$) and how to move forward (add $2$ to the previous term). This is a **recursive definition** — each term depends on the one before it. The rule $a_n = a_{n-1} + 2$ means we're building an arithmetic sequence, where the gap between consecutive terms is constant.

Why does this matter? Recognizing the pattern early lets us predict terms without grinding through the recursion every time. But let's build the first five terms carefully to see the structure emerge.

## Finding the first five terms

1. **Start with the given:** $a_1 = 1$.

2. **Apply the recursion for $n = 2$:**
   $$a_2 = a_1 + 2 = 1 + 2 = 3$$

3. **For $n = 3$:**
   $$a_3 = a_2 + 2 = 3 + 2 = 5$$

4. **For $n = 4$:**
   $$a_4 = a_3 + 2 = 5 + 2 = 7$$

5. **For $n = 5$:**
   $$a_5 = a_4 + 2 = 7 + 2 = 9$$

The first five terms are $1, 3, 5, 7, 9$ — the odd positive integers.

> [!TIP]
> Once you spot that each term increases by $2$, you can write the general term directly: $a_n = 1 + (n-1) \cdot 2 = 2n - 1$. This is the standard arithmetic sequence formula $a_n = a_1 + (n-1)d$ with $d = 2$.

## Writing the corresponding series …

Type	Formula	5th term
Linear	$a_n = 3n - 2$	$3(5)-2 = 13$
Quadratic	$a_n = n^2 + 1$	$5^2+1 = 26$
Exponential	$a_n = 2^n$	$2^5 = 32$
Rational	$a_n = \dfrac{n}{n+1}$	$\dfrac{5}{6}$
Alternating	$a_n = (-1)^n$	$(-1)^5 = -1$

Q.Let the sequence $a_n$ be defined as follows: $a_1 = 1$ , $a_n = a_{n-1} + 2$ for $n \geq 2$ . Find first five terms and write corresponding series.

Concept understanding — Sequence Terms Evaluation

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Finding the first five terms

Writing the corresponding series …

Q.Let the sequence ana_nan​ be defined as follows: a1=1a_1 = 1a1​=1, an=an−1+2a_n = a_{n-1} + 2an​=an−1​+2 for n≥2n \geq 2n≥2. Find first five terms and write corresponding series.

Finding the first five terms

Writing the corresponding series …

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Q.Let the sequence $a_n$ be defined as follows: $a_1 = 1$ , $a_n = a_{n-1} + 2$ for $n \geq 2$ . Find first five terms and write corresponding series.

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