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

Explanation:

Step1: Find the common difference (d)

The common difference (d=a_{2}-a_{1}). Given (a_{1} = 5) and (a_{2}=-14), then (d=-14 - 5=-19).

Step2: Use the formula for the (n)th term of an arithmetic sequence

The formula for the (n)th term of an arithmetic sequence is (a_{n}=a_{1}+(n - 1)d). Here (a_{1} = 5), (d=-19) and (n = 91). Substitute the values into the formula: (a_{91}=5+(91 - 1)\times(-19)). First, calculate ((91 - 1)=90). Then (90\times(-19)=-1710). Finally, (a_{91}=5-1710).

Answer:

(-1705)