Question 2

ThePerfectHacker · #1 $Old$ October 16th, 2006, 08:55 AM

A triangular number is the sequence:
1,3,6,10,15,21,...

A square number is the sequence:
1,4,9,16,25,...

Show that there are these two sequences contain infinitely many same numbers, i.e. 1 and 1.

topsquark · #2 $Old$ October 16th, 2006, 10:50 AM

Quote:

Originally Posted by ThePerfectHacker $View Post$

A triangular number is the sequence:
1,3,6,10,15,21,...

A square number is the sequence:
1,4,9,16,25,...

Show that there are these two sequences contain infinitely many same numbers, i.e. 1 and 1.

To clarify: You mean that the set of numbers that belong to both the triangular sequence and the square sequence has an infinite number of members?

-Dan

ThePerfectHacker · #3 $Old$ October 16th, 2006, 06:26 PM

Quote:

Originally Posted by topsquark $View Post$

To clarify: You mean that the set of numbers that belong to both the triangular sequence and the square sequence has an infinite number of members?

-Dan

Yes

ThePerfectHacker · #4 $Old$ October 17th, 2006, 07:42 AM

I do not understand the last line of the solution?

Soroban · #5 $Old$ October 17th, 2006, 08:26 AM

Hello, TPHacker!

Generally, with this type of problem we can generate solutions
with a recurrence formula of the form:

. . nth term .= .k·(preceding term) ± (term before) ... for some constant k.

In short, I "eyeballed" it . . .

ThePerfectHacker · #6 $Old$ October 17th, 2006, 09:23 AM

Beautiful job Soroban +rep+
Never seen it done like that.
Your own solution?

I have modified (not really the right word) your recurrence relation into a different form:
---
I always tried to show that the cubic sequence and triangular sequence only have a trivial solution i.e. 1. And it seems to be true for higher orders also but I was unable to prove that, you know how?

Soroban · #7 $Old$ October 17th, 2006, 02:37 PM

Hello again, TPHacker!

I got a different closed form for the recurrence.

It's not my own solution; I learned the method from a few of my books.
You may have seen something similar while exploring.

Now that LaTeX is back, I'll revise this post.

We have the recurrence: . $f(n) \:=\:6\cdot f(n-1) - f(n-2)$

Assume that $f(n)$ is exponential: . $f(n) \:=\:X^n$

The equation becomes: . $X^n \:=\:6X^{n-1} - X^{n-2}$

Divide by $X^{n-2}$ and we have: . $X^2 - 6X + 1 \:= \:0$

. . which has roots: . $X \:=\:3 \pm 2\sqrt{2}$

We form a linear combination of the two roots:
. . $f(n) \:=\:A(3 + 2\sqrt{2})^n + B(3 - 2\sqrt{2})^n$

To solve for $A$ and $B$ , we use the first two values of the sequence:

. . $\begin{array}{cc}f(1) \:= \\ f(2) \:=\end{array} \begin{array}{cc} A(3 + 2\sqrt{2}) + (3 - 2\sqrt{2})\\ A(3 + 2\sqrt{2})^2 + (3 - 2\sqrt{2})^2\end{array}\begin{array}{cc}= \:1 \\ = \:6\end{array}$

Solve the system and get: . $A \,= \,\frac{1}{4\sqrt{2}},\;\;B \,= \,-\frac{1}{4\sqrt{2}}$

Therefore: . $f(n) \;= \;\frac{(3 + 2\sqrt{2})^n - (3 - 2\sqrt{2})^n}{4\sqrt{2}}$

By the way: . $3 \pm 2\sqrt{2} \:=\:(1 \pm \sqrt{2})^2$

. . so you can rewrite the formula if you like . . .

ThePerfectHacker · #8 $Old$ October 23rd, 2006, 09:36 AM

Besides for Soroban's solution.... I have 3 more (1 of which I am too lazy to post).

Let $t_n$ be the $n-th$ triangular number.

Solution 1)
If t_n is a square then t_{4n(n+1)} is a square. How you are suppopsed to see that, I have no idea.

Solution 2)
Following the start of Soroban;s solution we arrive at the square root of 8k² +1, which is the "Pellian equation". Thus it has infinitely many solutions.