To find the 91st term of the arithmetic sequence $5, -14, -33, \dots$, let's analyze the problem step by step.

Step 1: Identify the First Term and Common Difference

1. The first term $a = 5$.
2. The common difference $d$ can be calculated as: $d = -14 - 5 = -19$

Step 2: Use the Formula for the $n$-th Term of an Arithmetic Sequence

The formula for the $n$-th term ($a_n$) of an arithmetic sequence is: $a_n = a + (n - 1) \cdot d$ where:

• $a$ is the first term,
• $d$ is the common difference, and
• $n$ is the term number.

Step 3: Substitute Values to Find the 91st Term

Here, $n = 91$, $a = 5$, and $d = -19$: $a_{91} = 5 + (91 - 1) \cdot (-19)$ $a_{91} = 5 + 90 \cdot (-19)$ $a_{91} = 5 - 1710$ $a_{91} = -1705$

Answer

The 91st term of the sequence is $-1705$.

Would you like a further breakdown or explanation?

Related Questions:

1. How would you find the sum of the first 91 terms of this arithmetic sequence?
2. What would be the 100th term in the same sequence?
3. How can you determine if a sequence is arithmetic or geometric?
4. If a term in this sequence is -195, what term number is it?
5. How would changing the common difference affect the sequence?

Tip: Remember, in an arithmetic sequence, the common difference is constant, which makes calculations straightforward with the formula for the $n$-th term.