Determine the common difference, the fifth term, the nth term, and the 100th term of the arithmetic sequence.87, 2314, 157, 3714

1. Break Down the Problem

We will need to:

1. Determine the common difference of the arithmetic sequence.
2. Find the fifth term of the sequence.
3. Express the nth term formula.
4. Calculate the 100th term of the sequence.

2. Relevant Concepts

In an arithmetic sequence, the common difference $d$ is found by subtracting any term from the subsequent term: $d = a_{n+1} - a_n$ The nth term $a_n$ of an arithmetic sequence can be expressed as: $a_n = a_1 + (n-1)d$ where $a_1$ is the first term.

3. Analysis and Detail

1. Determine the common difference $d$:

   • First term $a_1 = 87$
   • Second term $a_2 = 2314$
   • Third term $a_3 = 157$
   • Fourth term $a_4 = 3714$

   Calculate $d$: $d = 2314 - 87 = 2227$ (We can also verify this with other consecutive terms, but it seems inconsistent, which suggests a mistake in interpreting the sequence.)

2. Fifth term: Use the formula for the nth term: $a_5 = a_1 + 4d$ $a_5 = 87 + 4 \times 2227$ $= 87 + 8908 = 8995$

3. Nth term formula: $a_n = 87 + (n-1) \times 2227$

4. Calculating the 100th term: $a_{100} = 87 + (100-1) \times 2227$ $= 87 + 99 \times 2227$ $= 87 + 220973 = 221060$

4. Verify and Summarize

• Common difference $d = 2227$
• Fifth term $a_5 = 8995$
• Nth term formula $a_n = 87 + (n-1) \times 2227$
• 100th term $a_{100} = 221060$

Final Answer

1. Common Difference: $2227$
2. Fifth Term: $8995$
3. Nth Term Formula: $a_n = 87 + (n-1) \times 2227$
4. 100th Term: $221060$