Q.Write the first five terms of the following sequence and obtain the corresponding series: $a_1 = -1$, $a_n = \dfrac{a_{n-1}}{n}$, $n \geq 2$.

Concept: Sequence Terms Evaluation using a recursive formula.

We have $a_1 = -1$ and the recurrence $a_n = \dfrac{a_{n-1}}{n}$ for $n \geq 2$.

Step 1: Find $a_2 = \dfrac{a_1}{2} = \dfrac{-1}{2}$.

Step 2: Find $a_3 = \dfrac{a_2}{3} = \dfrac{-1/2}{3} = \dfrac{-1}{6}$.

Step 3: Find $a_4 = \dfrac{a_3}{4} = \dfrac{-1/6}{4} = \dfrac{-1}{24}$.

Step 4: Find $a_5 = \dfrac{a_4}{5} = \dfrac{-1/24}{5} = \dfrac{-1}{120}$.

The first five terms are $-1, -\dfrac{1}{2}, -\dfrac{1}{6}, -\dfrac{1}{24}, -\dfrac{1}{120}$. …

Understanding the Recursive Definition

A recursive sequence defines each term using previous terms. Here we're told that $a_1 = -1$ (the seed value) and every subsequent term follows the rule $a_n = \frac{a_{n-1}}{n}$ for $n \geq 2$. This means to find any term, we divide the previous term by the current position number.

The pattern will emerge quickly: each division by an increasing integer makes the terms shrink in absolute value, and since we start negative, all terms remain negative.

Computing the Terms

1. First term: Given directly as $a_1 = -1$.

2. Second term: Apply the recurrence with $n = 2$:

3. Third term: Now with $n = 3$:

4. Fourth term: With $n = 4$:

5. Fifth term: With $n = 5$:

The Corresponding Series

A series is formed by adding the terms of a sequence. The series corresponding to our sequence is: …