Determine the common difference, the fifth term, the nth term, and the 100th term of the arithmetic sequence.87, 2314, 157, 3714,   d= a5= an= a100=

1. Break Down the Problem

To solve the problem, we need to:

1. Determine the common difference $d$ of the arithmetic sequence.
2. Find the fifth term $a_5$.
3. Formulate the general term (nth term) $a_n$.
4. Calculate the 100th term $a_{100}$.

2. Relevant Concepts

In an arithmetic sequence:

• The common difference $d$ is calculated as: $d = a_{n} - a_{n-1}$
• The nth term can be expressed as: $a_n = a_1 + (n-1)d$
• The fifth term can be calculated using the formula for the nth term.

3. Analysis and Detail

1. First, we identify the first term $a_1 = 87$ and the second term $a_2 = 2314$.
2. Calculate the common difference $d$: $d = a_2 - a_1 = 2314 - 87 = 2227$
3. Calculate the fifth term $a_5$: Using the formula for the nth term: $a_5 = a_1 + (5-1)d = 87 + 4 \times 2227 = 87 + 8908 = 8995$
4. The general term $a_n$ is: $a_n = 87 + (n-1) \times 2227$
5. Calculate the 100th term $a_{100}$: $a_{100} = 87 + (100-1) \times 2227 = 87 + 99 \times 2227 = 87 + 219273 = 219360$

4. Verify and Summarize

• Common difference $d = 2227$
• Fifth term $a_5 = 8995$
• General term $a_n = 87 + (n-1) \times 2227$
• The 100th term $a_{100} = 219360$

Final Answer

• $d = 2227$
• $a_5 = 8995$
• $a_n = 87 + (n-1) \times 2227$
• $a_{100} = 219360$