Question

Find the 91st term of the arithmetic sequence 5, comma, minus, 14, comma, minus, 33, comma, dot dot dot5,−14,−33,...

Answer 1

To find the 91st term of the arithmetic sequence given by the terms 5, -14, -33, ..., we first identify the first term and the common difference of the sequence.

The first term \( a_1 \) is:
\[
a_1 = 5
\]

Next, we calculate the common difference \( d \) by subtracting the first term from the second term:
\[
d = a_2 - a_1 = -14 - 5 = -19
\]

Now we can use the formula for the \( n \)-th term of an arithmetic sequence, which is given by:
\[
a_n = a_1 + (n-1) \cdot d
\]

We need to find the 91st term (\( n = 91 \)):
\[
a_{91} = 5 + (91 - 1) \cdot (-19)
\]
\[
= 5 + 90 \cdot (-19)
\]
\[
= 5 - 1710
\]
\[
= -1705
\]

Thus, the 91st term of the arithmetic sequence is:
\[
\boxed{-1705}
\]