Question

1. determine if the sequence 83, 78, 73, 68, 63, … is an arithmetic sequence. if it is, then find the common difference.
2. what is the cost of renting the bicycle for 10 days? the explicit formula an = 14 + 12n gives the rental cost for n days. use the explicit formula to find the 10th term in the sequence.

Explanation:

1.

Step1: Check for common - difference

To determine if the sequence \(83,78,73,68,63,\cdots\) is arithmetic, find the difference between consecutive terms. Let \(a_1 = 83\), \(a_2=78\), \(a_3 = 73\).
\(a_2 - a_1=78 - 83=- 5\)
\(a_3 - a_2=73 - 78=-5\)

Since the difference between consecutive terms is constant (\(-5\)), the sequence is arithmetic.

Step1: Identify the formula and value of \(n\)

The explicit formula for the cost of renting a bicycle is \(a_n = 14+12n\), where \(n\) is the number of days. We want to find the cost for \(n = 10\) days.

Step2: Substitute \(n = 10\) into the formula

Substitute \(n = 10\) into \(a_n=14 + 12n\).
\(a_{10}=14+12\times10\)
\(a_{10}=14 + 120\)
\(a_{10}=134\)

Answer:

The sequence is arithmetic with a common - difference of \(-5\).

2.