Question

find the 91st term of the arithmetic sequence 5, -14, -33, ...

Explanation:

Step1: Identify first term $a_1$

$a_1 = 5$

Step2: Calculate common difference $d$

$d = -14 - 5 = -19$

Step3: Use arithmetic sequence formula

The formula for the $n$-th term of an arithmetic sequence is $a_n = a_1 + (n-1)d$. Substitute $n=91$, $a_1=5$, $d=-19$:

$$\begin{align*} a_{91}&= 5 + (91-1)(-19)\\ &= 5 + 90\times(-19) \end{align*}$$

Step4: Compute the final value

$$\begin{align*} a_{91}&= 5 - 1710\\ &= -1705 \end{align*}$$

Answer:

$-1705$