Question

which recursive sequence would produce the sequence 7, 37, 187, ...?
answer attempt 1 out of 2
\\(\circ\\) \\(a_1 = 7\\) and \\(a_n = 6a_{n-1} - 5\\)
\\(\circ\\) \\(a_1 = 7\\) and \\(a_n = -5a_{n-1} + 6\\)
\\(\circ\\) \\(a_1 = 7\\) and \\(a_n = 5a_{n-1} + 2\\)
\\(\circ\\) \\(a_1 = 7\\) and \\(a_n = 2a_{n-1} + 5\\)
submit answer

Explanation:

Step1: Test the first option

We know \(a_1 = 7\). For \(n = 2\), using the formula \(a_n=6a_{n - 1}-5\), we substitute \(n=2\) and \(a_1 = 7\) into it. So \(a_2=6\times a_1 - 5=6\times7-5 = 42 - 5=37\), which matches the second term. Now for \(n = 3\), \(a_3=6\times a_2-5=6\times37 - 5=222-5 = 217\)? Wait, no, the third term in the sequence is \(187\). Wait, maybe I made a mistake. Wait, let's check again. Wait, the sequence is \(7,37,187,\dots\). Wait, let's check the third term with the first option. Wait, \(a_1 = 7\), \(a_2=6\times7 - 5=37\), \(a_3=6\times37-5=222 - 5 = 217\), which is not \(187\). Wait, maybe I miscalculated. Wait, no, let's check the other options.

Wait, maybe I made a mistake. Let's check the third option. \(a_1 = 7\), \(a_n=5a_{n - 1}+2\). For \(n = 2\), \(a_2=5\times7+2=35 + 2=37\), which matches. For \(n = 3\), \(a_3=5\times37+2=185 + 2=187\), which matches the third term. Wait, but the first option gave \(a_3 = 217\), which is wrong. Wait, let's check the first option again. Wait, the first option: \(a_1 = 7\), \(a_n=6a_{n - 1}-5\). \(a_2=6\times7-5 = 37\), \(a_3=6\times37-5=222 - 5=217
eq187\). So first option is wrong.

Wait, let's check the third option: \(a_1 = 7\), \(a_n=5a_{n - 1}+2\). \(a_2=5\times7 + 2=37\), \(a_3=5\times37+2=185 + 2=187\), which matches the third term. Wait, but the first option's third term was wrong. Wait, maybe I made a mistake in the first option. Wait, no, the sequence is \(7,37,187\). Let's check the fourth option: \(a_n=2a_{n - 1}+5\). \(a_2=2\times7+5=14 + 5=19
eq37\), so fourth option is wrong. Second option: \(a_n=- 5a_{n - 1}+6\). \(a_2=-5\times7 + 6=-35 + 6=-29
eq37\), so second option is wrong. Third option: \(a_2=5\times7+2=37\), \(a_3=5\times37+2=187\), which matches. Wait, but the first option's \(a_3\) was \(217\), which is wrong. So the correct option is the third one: \(a_1 = 7\) and \(a_n=5a_{n - 1}+2\)? Wait, no, wait the options are:

1. \(a_1 = 7\) and \(a_n=6a_{n - 1}-5\)
2. \(a_1 = 7\) and \(a_n=-5a_{n - 1}+6\)
3. \(a_1 = 7\) and \(a_n=5a_{n - 1}+2\)
4. \(a_1 = 7\) and \(a_n=2a_{n - 1}+5\)

Wait, let's recalculate the first option. Wait, \(a_1 = 7\), \(a_2=6\times7-5 = 37\), \(a_3=6\times37-5=222 - 5=217\), but the sequence has \(187\) as the third term. So first option is wrong. Third option: \(a_1 = 7\), \(a_2=5\times7+2=37\), \(a_3=5\times37+2=185 + 2=187\), which matches. So the third option is correct? Wait, but the user's sequence is \(7,37,187\), so third option gives \(7,37,187\), which is correct. Wait, but let's check again.

Wait, maybe I made a mistake in the first option. Wait, no, \(6\times37 - 5=217\), which is not \(187\). So the third option is correct. Wait, but let's check the options again. The third option is \(a_1 = 7\) and \(a_n=5a_{n - 1}+2\). So that's the correct one.

Wait, but wait, the first option: let's check again. Wait, maybe the sequence is different. Wait, the problem says the sequence is \(7,37,187,\dots\). Let's check the third term with the first option: \(6\times37-5 = 217\), which is not \(187\). Third option: \(5\times37+2=187\), which is correct. So the correct option is the third one: \(a_1 = 7\) and \(a_n=5a_{n - 1}+2\).

Answer:

\(a_1 = 7\) and \(a_n = 5a_{n-1} + 2\) (the third option)