Use the explicit formula an=4−3n to find the seventh term of the arithmetic sequence.

Question

Use the explicit formula  an=4−3n  to find the seventh term of the arithmetic sequence.
Responses

a7=−1

a7=1

a7=−17

a7=7

GPT-4o mini · Answer

To find the seventh term $ a_7 $ of the arithmetic sequence given by the explicit formula $ a_n = 4 - 3n $, we can substitute $ n = 7 $ into the formula. $$ a_7 = 4 - 3(7) $$ Calculating this step-by-step: 1. Calculate $ 3(7) = 21 $ 2. Now substitute that back into the equation: $$ a_7 = 4 - 21 $$ 3. Finally, calculate $ 4 - 21 = -17 $ Thus, the seventh term of the arithmetic sequence is $$ \boxed{-17} $$

Yes. · Answer

Given the recursive formula for the geometric sequence a1=5, an=2/5an−1, find the second term of the sequence. Responses a2=2/5 a2=2 a2=8/5 a2=5 2/5

GPT-4o mini · Answer

Given the recursive formula for the geometric sequence $ a_1 = 5 $ and $ a_n = \frac{2}{5} a_{n-1} $, we need to find the second term $ a_2 $.

To find $ a_2 $, we substitute $ n = 2 $ into the recursive formula:

\[ a_2 = \frac{2}{5} a_1 \]

Since we already know that $ a_1 = 5 $:

\[ a_2 = \frac{2}{5} \times 5 \]

Calculating this:

\[ a_2 = \frac{2 \times 5}{5} = 2 \]

Thus, the second term of the sequence is

\[ \boxed{2} \]

Yes. · Answer

Use the recursively defined geometric sequence a1=5/6, an=4an−1 and find the common ratio. Responses 56 −23 103 4

GPT-4o mini · Answer

In a geometric sequence, the common ratio $ r $ is defined as the factor by which we multiply one term to get the next term. Given the recursive formula for the geometric sequence:

$ a_1 = \frac{5}{6} $
$ a_n = 4 a_{n-1} $

we can see that to go from term $ a_{n-1} $ to term $ a_n $, we multiply by 4. Thus, the common ratio $ r $ is:

\[ r = 4 \]

Therefore, the common ratio of the sequence is

\[ \boxed{4} \]

Yes. · Answer

Find the recursive formula for the arithmetic sequence 7, 5, 3, 1, . . . Responses a1=7, an=2an−1 a1=7, an=an−1−2 a1=7, an=−2an−1 a1=7, an=an−1+2

GPT-4o mini · Answer

To find the recursive formula for the arithmetic sequence $ 7, 5, 3, 1, \ldots $, we first need to identify the first term and the common difference.

The first term $ a_1 $ is given as $ 7 $.
The common difference $ d $ between consecutive terms can be calculated as follows:
- $ 5 - 7 = -2 $
- $ 3 - 5 = -2 $
- $ 1 - 3 = -2 $

The common difference $ d $ is $ -2 $.

In a recursive formula for an arithmetic sequence, $ a_n $ can be defined in terms of the previous term as:

\[ a_n = a_{n-1} + d \]

Since $ d = -2 $, the formula becomes:

\[ a_n = a_{n-1} - 2 \]

Therefore, the recursive formula for the sequence is:

\[ a_1 = 7, \quad a_n = a_{n-1} - 2 \]

The correct option is:

\[ \boxed{a_1 = 7, , a_n = a_{n-1} - 2} \]

Yes. · Answer

Which of the following correctly uses the explicit formula, An=1/4+(n−1)⋅1/2 , to find the 9th term of the described arithmetic sequence? Responses 4 1/4 8 1/4 4 3/4 8 3/4

GPT-4o mini · Answer

To find the 9th term of the arithmetic sequence using the explicit formula given by

\[ A_n = \frac{1}{4} + (n - 1) \cdot \frac{1}{2} \]

we will substitute $ n = 9 $ into the formula.

\[ A_9 = \frac{1}{4} + (9 - 1) \cdot \frac{1}{2} \]

Calculating step-by-step:

Calculate $ 9 - 1 = 8 $.
Now substitute that back into the equation: \[ A_9 = \frac{1}{4} + 8 \cdot \frac{1}{2} \]
Calculate $ 8 \cdot \frac{1}{2} = 4 $.
Now substitute that into the equation: \[ A_9 = \frac{1}{4} + 4 \]
To add these, convert $ 4 $ to a fraction: \[ 4 = \frac{16}{4} \]
Now add: \[ A_9 = \frac{1}{4} + \frac{16}{4} = \frac{17}{4} \]

Now, converting $ \frac{17}{4} $ to a mixed number gives us $ 4 \frac{1}{4} $.

Therefore, the 9th term of the arithmetic sequence is

\[ \boxed{4 \frac{1}{4}} \]