# Nth term of a sequence

1. Sep 26, 2010

### zorro

1. The problem statement, all variables and given/known data
Is this statement correct?

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

2. Relevant equations

3. 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?

2. Sep 26, 2010

### Staff: Mentor

The book's statement doesn't seem correct to me. The simplest sequence that fits the description is an = {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.

3. Sep 26, 2010

### zorro

Thanks!

4. Sep 26, 2010

### Office_Shredder

Staff Emeritus
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

5. Sep 26, 2010

### 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.

6. Sep 26, 2010

### Office_Shredder

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

7. Sep 26, 2010

### 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.

8. Sep 26, 2010

### Office_Shredder

Staff Emeritus
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

9. Sep 27, 2010

### 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.

10. Sep 27, 2010

### zorro

Can you give an example?

11. Sep 27, 2010

### Staff: Mentor

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

12. Sep 27, 2010

Thanks!