Sequence

Algebra Level 3

Consider the following sequences:

(1) $1,3,5,7,9,...$

(2) the arithmetic sequence $2,5,8,11,14, ...$

(3) the geometric sequence $3,9,27,81, ...$

In how many of the above sequences have we precisely defined the $6^\text{th}$ term?

0 1 2 3

3 solutions

Margaret Zheng
Apr 9, 2016

Relevant wiki: Simple Sequences

Hmm. It looks like we are able to determine the 6th term of all three of those sequences, right? NO.

The problem is, the first sequence does not give a "pattern". For the second sequence, it is given that the sequence is arithmetic; therefore, we can determine from the sequence that its common difference is $3$ and the first term is $2$ . Similar for the third sequence, the common ratio is $3$ and the first term is $3$ .

However, we have no clue how the first sequence is going to develop. You might say that it is going to be $11$ . Well, why couldn't it be 31, 51, 17403 or 12394869843595479713459? It is a sequence that does not have a known definition, so anyone could define the 6th term however they like.

Remember, by only showing the first few terms of a sequence is not called "defining" a sequence!

I did not get the solution and the question, could you please elaborate more

Syed Baqir - 5 years, 2 months ago

I am not able to understand how 31,51 comes in the sequence of 1, 3, 5, 7,9,

Arun Krishna AMS - The Joker - 5 years, 1 month ago

What I was trying to say is that if the sequence is not well defined, the next term could literally be anything. (quote from @William Pan : remember not to assume anything!) However, (for example) the second sequence says "the arithmetic sequence" and by looking at the sequence, we know that its common difference is 3 and the first term is 2.

Margaret Zheng - 5 years, 1 month ago

It can be $1,3,5,7,9,31$ . This is a sequence for the values of $f(x)=\dfrac{7}{240}x^6-\dfrac{107}{240}x^5+\dfrac{125}{48}x^4-\dfrac{349}{48}x^3+\dfrac{148}{15}x^2-\dfrac{227}{60}x$ for $x=1,2,3,4,5,6,7,8,\ldots$ Your 7th term will be $154$

Similarly, you can have $51,71,17403$ or whatever number you want as the 6th term. I have a few more examples in my solution below

Hung Woei Neoh - 5 years ago

how did you get the polynomial?

Terrell Bombb - 4 years, 6 months ago

please help me understand ... I cannot get it how the first one only does not have a pattern while the others have....How to know which sequence shows which pattern?

Tanmoy Dutta - 4 years, 8 months ago

Hung Woei Neoh
Jun 14, 2016

The first sequence has no specific description. Therefore, there are many possibilities that we can use. The most obvious one is:

Arithmetic progression. The sequence will be $1,3,5,7,9,11,\ldots$

However, note that I can define a sequence like this:

Odd positive integers whose digits are not repeated, arranged in ascending order. This gives: $1,3,5,7,9,13,15,\ldots$
Odd positive integers whose digits alternate between even and odd. This gives: $1,3,5,7,9,21,23,\ldots$
The values of $f(x)=-\dfrac{1}{120}x^5 +\dfrac{1}{8}x^4-\dfrac{17}{24}x^3 +\dfrac{15}{8}x^2-\dfrac{17}{60}x$ for $x=1,2,3,4,5,6,7,\ldots$ This gives $1,3,5,7,9,10,7,\ldots$

Therefore, for sequence 1, we do not have a precise definition for the 6th term.

The second and third sequence, however, have precisely defined the 6th term. The second sequence is an AP with $a=2,d=3$ . The 6th term is $17$ . The third sequence is a GP with $a=3,r=3$ . The 6th term is $729$

Therefore, only $\boxed{2}$ sequences have a precise definition for the 6th term

why is the 1st sequence not defined i could not understand?

Karan Pattanaik - 2 years, 7 months ago

because there are multiple ways/functions we can predict the 6th term,so the sixth term cant be precisely predicted. The point here is the sequence has satisfied multiple functions for all the terms in the sequence and we need only one precise function

ulrich henriques - 9 months ago

@Hung Woei Neoh how did you get this function ?

Rania El Hagin 10 - 2 years, 2 months ago

i guess he has use wolfram alpha

ulrich henriques - 9 months ago

William Pan
Apr 9, 2016

We don't know what kind of sequence number one is. Therefore, we cannot determine the sixth term of the first sequence.

Remember to not assume anything.

why is the 1st sequence not defined ?

Karan Pattanaik - 2 years, 7 months ago

The first sequence to me, personally, was a arithmetic sequence because it contains a constant. Which is two.

Queen Slayy - 2 years, 5 months ago

It's not clear

Gaurav Bhatt - 1 year, 2 months ago

Yeah!it is confusing

Kelly Quinn - 4 days, 22 hours ago

Sequence

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