n-th term of this sequence

M.RezaMathing

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Hello I hope you are having a great day. I would be very glad if you answered my question. Thanks in advance.

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?
 

Subhotosh Khan

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Hello I hope you are having a great day. I would be very glad if you answered my question. Thanks in advance.

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

a1 = 5

a2 = a(1) *7 + 5 = a(2-1) *7 + 5

a3 = a(2) *7 + 5 = a(3-1) * 7 + 5

a4 = a(3) *7 + 5 = a(4-1) *7 + 5
.
.
.
an = ??
 

Dr.Peterson

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Hello I hope you are having a great day. I would be very glad if you answered my question. Thanks in advance.

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?
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:

 

Dr.Peterson

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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.

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'll tell you specifically what I did (looking for something you could do with no special knowledge).

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.
 

HallsofIvy

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When I was a freshman in college a math professor wrote the sequence 85, 84, 83 on the board and asked for the next number in the sequence. Of course everyone answered "82". NO, the next number in the sequence was 29! These were the numbers of the subway stops he passed on his way to work. At at 83 the train changed to a different line. His point was that any finite sequence has infinitely many continuations. The pattern, if there IS a pattern, does not have to be simple!
 

Dr.Peterson

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When I was a freshman in college a math professor wrote the sequence 85, 84, 83 on the board and asked for the next number in the sequence. Of course everyone answered "82". NO, the next number in the sequence was 29! These were the numbers of the subway stops he passed on his way to work. At at 83 the train changed to a different line. His point was that any finite sequence has infinitely many continuations. The pattern, if there IS a pattern, does not have to be simple!
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.
 

tkhunny

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To Emphasize: Any FINITE number of terms has INfINITELY many solutions for the "NextTerm".

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 ALL SUCH QUESTIONS WIth "-2". This person was intelligent based on any criteria with which I am familiar,

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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