Q.Write the first five terms of the following sequence and obtain the corresponding series: $a_1 = a_2 = 2$, $a_n = a_{n-1} - 1$, $n > 2$.

The key idea is to evaluate the sequence terms one by one using the given recurrence relation, starting from the initial conditions.

We are given $a_1 = 2$ and $a_2 = 2$. For $n > 2$, the rule is $a_n = a_{n-1} - 1$.

• For $n = 3$: $a_3 = a_2 - 1 = 2 - 1 = 1$.
• For $n = 4$: $a_4 = a_3 - 1 = 1 - 1 = 0$.
• For $n = 5$: $a_5 = a_4 - 1 = 0 - 1 = -1$. …

Why This Approach Works

When a sequence is defined recursively, you cannot jump ahead — you must build it term by term, starting from the given initial values. Here, the rule $a_n = a_{n-1} - 1$ only applies for $n > 2$, meaning the first two terms are special. The key insight: after the second term, the sequence becomes an arithmetic progression with common difference $-1$, but the first two terms are identical, so the "drop" happens only from $a_2$ to $a_3$.

The series simply means the sum of the terms you list. So we first generate the terms, then add them.

Step-by-Step Solution

1. Identify the given initial terms.

   The problem states $a_1 = 2$ and $a_2 = 2$. These are fixed and do not follow the recurrence rule.

2. Apply the recurrence for $n = 3$.

   For $n > 2$, we use $a_n = a_{n-1} - 1$.

   So $a_3 = a_2 - 1 = 2 - 1 = 1$.

3. Continue for $n = 4$.

   $a_4 = a_3 - 1 = 1 - 1 = 0$.

4. Continue for $n = 5$.

   $a_5 = a_4 - 1 = 0 - 1 = -1$. …