#### M.RezaMathing

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Whats is the n-th term of this sequence:

**5, 40, 285, 2000, ...**

I think each term is multiplied by 7 then 5 is added to it.

Also what is the general formula for this type of sequences?

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- Thread starter M.RezaMathing
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Whats is the n-th term of this sequence:

I think each term is multiplied by 7 then 5 is added to it.

Also what is the general formula for this type of sequences?

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Great observation. That is correct - for the four numbers shown. We are going to assume that the pattern continues. Then we write:

Whats is the n-th term of this sequence:

5, 40, 285, 2000, ...

I think each term is multiplied by 7 then 5 is added to it.

Also what is the general formula for this type of sequences?

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We can never be certain of a formula when we are only given a few terms, but the recursive formula (also called a recurrence relation) that you have guessed seems reasonable.

Whats is the n-th term of this sequence:

5, 40, 285, 2000, ...

I think each term is multiplied by 7 then 5 is added to it.

Also what is the general formula for this type of sequences?

There are various ways to turn a recurrence into an explicit formula (in easy cases), depending on what you have learned. Here is a lesson that includes an example like yours:

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There seems to be only one with these terms.

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I'll tell you specifically what I did (looking for something you could do with no special knowledge).We can never be certain of a formula when we are only given a few terms, but the recursive formula (also called a recurrence relation) that you have guessed seems reasonable.

There are various ways to turn a recurrence into an explicit formula (in easy cases), depending on what you have learned. Here is a lesson that includes an example like yours:

I recognized that, with repeated multiplication by 7, the sequence should be related to 7^n. I listed the terms of both sequences and compared them, observed a relationship between them, and wrote an equation I could solve. Then I confirmed that my (largely guessed) formula satisfied the recurrence and gave the correct first four terms.

The method shown in the page I referred to takes more work but less guessing.

But pka's OEIS entry implies a beautiful shortcut: The pattern is obvious in base 7, which leads to a simple geometric series formula that agrees with what I got.

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Here's an even better example of the same sort. I quote from a comment by one of my colleagues:

These sorts of patterns are used in intelligence tests, and the "correct" answer is "whatever very intelligent people think the correct answer is". That's not much help, is it?

I remember a wonderful example shown to me once that illustrated how silly this sort of question is. Here it is:

What comes next in this sequence?33, 23, 14, 9, ___

The answer is "Christopher Street". The reason is that the numbers are the exits of the 6th Avenue subway in New York City.

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If this kind of problem serves any purpose, I have never been able to tell what it is. Some say "Pattern Recognition" is an important skill. I don't disagree with that, but the inappropriate application of an arbitrary pattern is NOT an important skill.

I knew an individual who answered

This kind of question, if it has an answer, falls under the question type: "Guess what I have floating around in my head?" It is not a worthwhile question type. It should die a quick and perhaps miserable death.

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