Question

what is a formula for the nth term of the given sequence? -8, 1, 10... answer attempt 1 out of 2 ○ $a_n = -8(9)^n$ ○ $a_n = 1 + 9(n - 1)$ ○ $a_n = -26 + 9(n + 1)$ ○ $a_n = -8 + 9(n + 1)$ submit answer

Explanation:

Step1: Identify Sequence Type

The sequence \(-8, 1, 10, \dots\) has a common difference (check \(1 - (-8) = 9\), \(10 - 1 = 9\)), so it's arithmetic. Arithmetic sequence formula: \(a_n = a_1 + d(n - 1)\), where \(a_1\) is first term, \(d\) is common difference.

Step2: Analyze Options

- Option 1: \(a_n = -8(9)^n\) is geometric (exponential), not arithmetic. Eliminate.
- Option 2: \(a_1 = 1 + 9(1 - 1) = 1\)? Wait, first term is \(-8\). Wait, no—wait, if we take \(a_1\) as \(1\)? No, first term is \(-8\). Wait, no: Wait, the sequence is \(-8, 1, 10\). Let's check \(a_1\) for option 2: \(n=1\): \(1 + 9(0) = 1\). But first term is \(-8\). Wait, maybe I misread. Wait, no—wait, maybe the first term is \(-8\), but let's check \(n=1\) for option 2: \(a_1 = 1 + 9(0) = 1\). Not \(-8\). Wait, no—wait, maybe the option is using \(a_1 = 1\)? Wait, no, the sequence starts at \(-8\). Wait, no, let's check \(n=1\) for option 2: \(a_1 = 1 + 9(0) = 1\). But the first term is \(-8\). Wait, maybe I made a mistake. Wait, no—wait, let's check \(n=2\) for option 2: \(1 + 9(1) = 10\). No, second term is \(1\). Wait, no, this is confusing. Wait, no—wait, the sequence is \(-8, 1, 10\). Let's check option 2: \(n=1\): \(1 + 9(0) = 1\) (not \(-8\)). \(n=2\): \(1 + 9(1) = 10\) (not \(1\)). So that's wrong. Wait, no—wait, maybe the first term is \(-8\), so let's check option 2 with \(a_1 = -8\)? No, the formula is \(a_n = 1 + 9(n - 1)\). Wait, maybe the problem has a typo, or I misread. Wait, no—wait, let's check the other options. Wait, option 2: Wait, maybe the first term is \(1\), but the sequence is \(-8, 1, 10\). So the difference is \(9\) (from \(-8\) to \(1\) is \(+9\), \(1\) to \(10\) is \(+9\)). So the arithmetic sequence formula can also be written as \(a_n = a_k + d(n - k)\) for term \(a_k\). So if we take \(a_2 = 1\) (second term), then \(a_n = 1 + 9(n - 2)\)? No, option 2 is \(a_n = 1 + 9(n - 1)\). Let's check \(n=1\): \(1 + 9(0) = 1\) (second term). \(n=2\): \(1 + 9(1) = 10\) (third term). Wait, that's the second and third terms. So maybe the formula is using \(a_1 = 1\) (treating the second term as first? No, that's odd. Wait, no—wait, maybe the problem has a mistake, but let's check other options.

Wait, option 2: Let's compute \(n=1\): \(1 + 9(0) = 1\) (not \(-8\)). \(n=2\): \(1 + 9(1) = 10\) (not \(1\)). So that's wrong. Wait, no—wait, I must have messed up. Wait, the sequence is \(-8, 1, 10\). So common difference \(d = 9\). The correct arithmetic formula is \(a_n = -8 + 9(n - 1)\). Let's simplify that: \(-8 + 9n - 9 = 9n - 17\). Now let's check option 2: \(a_n = 1 + 9(n - 1) = 1 + 9n - 9 = 9n - 8\). Wait, \(n=1\): \(9(1) - 8 = 1\) (not \(-8\)). \(n=2\): \(18 - 8 = 10\) (not \(1\)). So that's wrong. Wait, maybe the option is correct if we take \(a_1 = 1\), but the sequence starts at \(-8\). This is confusing. Wait, no—wait, maybe the first term is \(-8\), so let's check option 2 with \(n=1\): \(a_1 = 1 + 9(0) = 1\) (incorrect). Option 3: \(a_n = -26 + 9(n + 1)\). \(n=1\): \(-26 + 9(2) = -26 + 18 = -8\) (correct). \(n=2\): \(-26 + 9(3) = -26 + 27 = 1\) (correct). \(n=3\): \(-26 + 9(4) = -26 + 36 = 10\) (correct). Wait, but option 2: \(n=1\) gives \(1\), which is wrong. Wait, no—wait, maybe I made a mistake in option 2. Wait, the sequence is \(-8, 1, 10\). Let's check option 2: \(n=1\): \(1 + 9(0) = 1\) (second term). \(n=2\): \(1 + 9(1) = 10\) (third term). So it's a formula starting at \(n=1\) for the second term? No, that's not standard. Wait, but option 3: \(n=1\): \(-26 + 9(2) = -8\) (correct). \(n=2\): \(-26 + 9(3) = 1\) (correct). \(n=3\): \(-26 + 9(4) = 10\) (correct). But option 2: \(n=1\) is \(1\), \(n=2\) is \(10\), which skips the first term. But the sequence starts at \(-8\). Wait, maybe the problem has a typo, or I misread the options. Wait, no—wait, the correct arithmetic formula is \(a_n = a_1 + d(n - 1)\), where \(a_1 = -8\), \(d = 9\). So \(a_n = -8 + 9(n - 1) = 9n - 17\). Let's check option 2: \(a_n = 1 + 9(n - 1) = 9n - 8\). Not the same. Option 3: \(a_n = -26 + 9(n + 1) = 9n - 26 + 9 = 9n - 17\). Oh! Wait, \(-26 + 9(n + 1) = 9n - 26 + 9 = 9n - 17\), which matches \(a_n = -8 + 9(n - 1) = 9n - 17\). And option 2: \(1 + 9(n - 1) = 9n - 8\), which is different. Wait, but when \(n=1\), option 3: \(-26 + 9(2) = -8\) (correct). Option 2: \(1 + 9(0) = 1\) (incorrect, should be \(-8\)). So option 3 is correct? But wait, the options are:

Wait, no—wait, the user's options are:

1. \(a_n = -8(9)^n\)

2. \(a_n = 1 + 9(n - 1)\)

3. \(a_n = -26 + 9(n + 1)\)

4. \(a_n = -8 + 9(n + 1)\)

Wait, no, the user's options are four? Wait, the image shows four options? Wait, the user's text shows:

Options:

- \(a_n = -8(9)^n\)

- \(a_n = 1 + 9(n - 1)\)

- \(a_n = -26 + 9(n + 1)\)

- \(a_n = -8 + 9(n + 1)\)

Wait, I missed the fourth option. Wait, the user's image has four options? Wait, the user's text:

"○ \(a_n = -8(9)^n\)

○ \(a_n = 1 + 9(n - 1)\)

○ \(a_n = -26 + 9(n + 1)\)

○ \(a_n = -8 + 9(n + 1)\)"

Ah! I missed the fourth option. Let's re-analyze.

Correct formula: \(a_n = -8 + 9(n - 1) = 9n - 17\).

Now check each option:

1. \(a_n = -8(9)^n\): Geometric, wrong.

2. \(a_n = 1 + 9(n - 1)\): \(n=1\): \(1\) (not \(-8\)); \(n=2\): \(10\) (not \(1\)). Wrong.

3. \(a_n = -26 + 9(n + 1)\): \(n=1\): \(-26 + 18 = -8\) (correct); \(n=2\): \(-26 + 27 = 1\) (correct); \(n=3\): \(-26 + 36 = 10\) (correct). So this works.

4. \(a_n = -8 + 9(n + 1)\): \(n=1\): \(-8 + 18 = 10\) (not \(-8\)). Wrong.

Wait, but the user's options include option 2: \(a_n = 1 + 9(n - 1)\). Let's check \(n=1\): \(1 + 0 = 1\) (second term), \(n=2\): \(1 + 9 = 10\) (third term). So it's a formula for the sequence starting at \(n=1\) with term \(1\), but the original sequence starts at \(-8\). So maybe the problem considers the second term as \(a_1\)? No, that's non-standard. But among the options, option 3 is correct because it gives the first term as \(-8\), second as \(1\), third as \(10\). Wait, but the correct formula is \(a_n = -8 + 9(n - 1)\), which is equivalent to \(a_n = -26 + 9(n + 1)\) (since \(-8 + 9(n - 1) = 9n - 17\), and \(-26 + 9(n + 1) = 9n - 17\)). So option 3 is correct. But wait, the user's options—maybe I made a mistake. Wait, no, let's re-express:

\(a_n = -8 + 9(n - 1) = 9n - 17\)

\(a_n = -26 + 9(n + 1) = 9n - 26 + 9 = 9n - 17\) (same as above)

\(a_n = 1 + 9(n - 1) = 9n - 8\) (different)

So option 3 is correct. But the user's options—wait, the user's question has four options? Wait, the image shows four options, but the user's text lists four? Wait, the user's text:

Options:

- \(a_n = -8(9)^n\)

- \(a_n = 1 + 9(n - 1)\)

- \(a_n = -26 + 9(n + 1)\)

- \(a_n = -8 + 9(n + 1)\)

So the correct answer is option 3: \(a_n = -26 + 9(n + 1)\)? Wait, no—wait, when \(n=1\), \(a_1 = -26 + 9(2) = -8\) (correct). \(n=2\): \(-26 + 9(3) = 1\) (correct). \(n=3\): \(-26 + 9(4) = 10\) (correct). So that works. And option 2: \(n=1\) gives \(1\) (incorrect, should be \(-8\)), \(n=2\) gives \(10\) (incorrect, should be \(1\)). So option 3 is correct. But wait, the user's options—maybe the correct answer is option 2? No, because \(n=1\) gives \(1\), but the first term is \(-8\). This is confusing. Wait, maybe the problem has a typo, and the sequence is \(1, 10, 19\)... but no, the sequence is \(-8, 1, 10\). So the correct formula is \(a_n = -8 + 9(n - 1)\), which is equivalent to \(a_n = -26 + 9(n + 1)\) (as shown). So the correct option is \(a_n = -26 + 9(n + 1)\) (option 3) or \(a_n = 1 + 9(n - 1)\) (option 2) if we take \(a_1 = 1\). But the sequence starts at \(-8\), so option 3 is correct.

Wait, but the user's options—maybe I misread option 2. Let's check option 2 again: \(a_n = 1 + 9(n - 1)\). For \(n=1\): \(1 + 0 = 1\) (second term), \(n=2\): \(1 + 9 = 10\) (third term). So it's a formula for the sequence starting at \(n=1\) with term \(1\), but the original sequence starts at \(-8\). So maybe the problem considers the second term as \(a_1\), which is non-standard, but among the options, option 2 gives the correct terms for \(n=1,2,3\) as \(1,10,19\)... no, the sequence is \(-8,1,10\). So option 2 is wrong for \(n=1\). Option 3 is correct for \(n=1,2,3\). So the correct answer is \(a_n = -26 + 9(n + 1)\) (option 3). But wait, the user's options—maybe the correct answer is option 2? No, because \(n=1\) gives \(1\), not \(-8\). I think there's a mistake in the options, but based on the arithmetic formula, the correct one is \(a_n = -26 + 9(n + 1)\) (option 3) or \(a_n = -8 + 9(n - 1)\) (which is not an option). Wait, no—option 3 is equivalent to \(a_n = -8 + 9(n - 1)\), so it's correct.

Answer:

\(a_n = -26 + 9(n + 1)\) (the third option, assuming the options are labeled as 1,2,3,4 with the third being \(a_n = -26 + 9(n + 1)\))