Proof: Quadratic Nth Term of Sequence Indicates Arithmetic Progression

[zorro]

Homework Statement

Is this statement correct?

If the nth term of a sequence is a quadratic expression in n, then the sequence is an A.P.

Homework Equations

The Attempt at a Solution

Take arbitrary t(n)=n^2-2n-2
I substituted 1,2,3 in the above expression and noted the c.d.
It is not constant.
But the book says that this statement is correct.
Any ideas?

 

[Mark44]

The book's statement doesn't seem correct to me. The simplest sequence that fits the description is a_n = {n²} = {1, 4, 9, 16, ..., n², ...} This is definitely not an arithmetic sequence for the reason you stated - the difference between pairs of successive terms is not constant.

 

[zorro]

Thanks!

 

[Office_Shredder]

If the terms of a sequence are from a quadratic formula, then the difference between the nth and (n-1)st terms form an arithmetic progression.

For example if the sequence is 1,4,9,16,25,...

then the differences are

4-1, 9-4, 16-9, 25-16,...
3,5,7,9,...

That might be what they meant to refer to

 

[zorro]

It is true only in case on n^2 (may be in some other cases too).
But if you take an expression like n^2 +2n-1, then Tn- T(n-1)=2n is not independent of n i.e. it is not a constant.

 

[Office_Shredder]

The sequence whose nth term is 2n is an arithmetic progression

 

[zorro]

You did not get me. In my expression, 2n is the difference between two consecutive terms of the sequence. It is not the nth term.
If you take nth term of the sequence as 2n, it violates the question as it is not a quadratic expression.

 

[Office_Shredder]

What I said is that the sequence whose nth term is the difference between consecutive terms of the quadratic sequence is an arithmetic progression. So for your example, the nth term of the sequence I'm describing is T(n)-T(n-1), and this new sequence is an arithmetic progression

 

[HallsofIvy]

If the sum (form i= 1 to n) of a sequence of numbers is a quadratic function of n, then the sequence is arthmetic.

 

[zorro]

Can you give an example?

 

[Mark44]

If the sum (form i= 1 to n) of a sequence of numbers is a quadratic function of n, then the sequence is arthmetic.

Can you give an example?

The sum of the first n integers.
\sum_{k= 1}^n k = \frac{n(n + 1)}{2}

 

[zorro]

Thanks!