Question

Which of the following recursive formulas represent the same arithmetic sequence as the explicit formula an=5+(n-1)2

Answer 1

Answer:

Choice A:

$a_1=5$

$a_{n}=a_{n-1}+2$

Step-by-step explanation:

$a_n=5+(n-1)2$

means we looking for first term 5 and the sequence is going up by 2.

In general,

$a_n=a_1+(n-1)d$

means you have first term $a_1$ and the sequence has a common difference of d.

So it is between the first two choices.

The explicit form of an arithmetic sequence is: $a_n=a_1+(n-1)d$

An equivalent recursive form is $a_n=a_{n-1}+d \text{ where } a_1 \text{ is the first term}$

So d again here is 2.

So choice a is correct.

$a_1=5$

$a_{n}=a_{n-1}+2$

Answer 2

Answer: Option a

$\left \{ {{a_1=5} \atop {a_n=a_{(n-1)}+2}} \right.$

Step-by-step explanation:

The arithmetic sequences have the following explicit formula

$a_n=a_1 +(n-1)*d$

Where d is the common difference between the consecutive terms and $a_1$ is the first term of the sequence:

The recursive formula for an arithmetic sequence is as follows

$\left \{ {{a_1} \atop {a_n=a_{(n-1)}+d}} \right.$

Where d is the common difference between the consecutive terms and $a_1$ is the first term of the sequence:

In this case we have the explicit formula $a_n=5+(n-1)*2$

Notice that in this case

$a_1 = 5\\d = 2$

Then the recursive formula is:

$\left \{ {{a_1=5} \atop {a_n=a_{(n-1)}+2}} \right.$

The answer is the option a.