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If you know the fifth term in an arithmetic sequence and the common difference, can you write a recursive formula for the sequence? Explain why or why not.

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Respuesta :

DonSmilo
If the arithmetic sequence is linear then yes, you can find the sequence with the common difference.
For example, if the common difference is add one now and you know that the fifth term is the number six, then you would know that the sequence is two, three, four, five, six.

to work out the sequence going backwards from the fifth term, you just need to subtract the amount that was previously added. For example, if the sequence adds three every time you go to the next term, then when figuring out the previous term just subtract three from the current term. So to figure out the fourth term, subtract three from the fifth term.

facundo3141592

Given that we know the fifth term in an arithmetic sequence and the common difference, we want to see if we can get the recursive formula for the sequence. We will see that it is enough to completely describe the sequence.

We do know that the general recursive formula is given by:

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

Where d is the common difference.

So if we know the fifth term, we know:

[tex]a_5[/tex]

And we know the value of d.

Then the first term of the sequence

[tex]a_1 = a_5 - 4*d[/tex]

Thus the n-th term of the arithmetic sequence will be given by:

[tex]a_n = (a_5 - 4*d) + n*d = a_5 + (n - 4)*d[/tex]

So now we can get every term of the sequence.

Thus we proved that knowing one term and the common difference is enough to completely describe the arithmetic sequence.

If you want to learn more, you can read:

https://brainly.com/question/18109692

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