Question

write an expression to describe the sequence below, and then find the 82nd term. use n to represent the position of a term in the sequence, where n = 1 for the first term. 85, 170, 255, 340, ... a_n = a_82 =

Explanation:

Step1: Identify sequence type

The sequence 85, 170, 255, 340,... has a common difference. Check: \(170 - 85 = 85\), \(255 - 170 = 85\), \(340 - 255 = 85\). So it's an arithmetic sequence with first term \(a_1 = 85\) and common difference \(d = 85\).

Step2: Arithmetic sequence formula

The formula for the \(n\)-th term of an arithmetic sequence is \(a_n = a_1 + (n - 1)d\). Substitute \(a_1 = 85\) and \(d = 85\):
\[

$$\begin{align*} a_n &= 85 + (n - 1) \times 85\\ &= 85 + 85n - 85\\ &= 85n \end{align*}$$

\]

Step3: Find the 82nd term

Substitute \(n = 82\) into \(a_n = 85n\):
\(a_{82} = 85 \times 82 = 6970\)

Answer:

\(a_n = 85n\)
\(a_{82} = 6970\)