New to Forum: Proving f(n+3) -3f(n+2) + 3f(n+1)-f(n) = 0

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In summary, the conversation discusses the problem of showing that the equation f(n+3) -3f(n+2) + 3f(n+1)-f(n) = 0 holds for all arithmetic progressions. The participants suggest using a simple arithmetic progression to test the hypothesis and provide a formula for the sum of n terms of an arithmetic progression. They also suggest calculating specific values to plug into the equation to prove its validity.
  • #1
Hi people, as you can see i am new in this forum.

Sorry if i am posting in wrong section.

Im going to try to traduce the problem.

" f(n) is the sum o n terms of a arithmetic progression. "
Show that :

f(n+3) -3f(n+2) + 3f(n+1)-f(n) = 0

I just want to know if

-3f(n+2) = f(-3n -6) ??
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  • #2
Hi Fabio,

couldn't you test this yourself? Why don't you define an arithmetic progression and try it out to see if it works?

EDIT: You could use the simplest arithmetic progression, f(n) = n
  • #3
Oh, so easy.

Thanks, i never thought that i could define the arithmetic progression.
  • #4
Well, what dacruick means is that you can use that particular sequence to "test" your hypothesis that -3f(n+2)= f(-3n- 6). In fact, since f(n) is defined as "the sum of the first n terms of an arithmetic sequence", I can't help but wonder what "f(-3n-6)" means! Of course, that would not prove the theorem that
f(n+3) -3f(n+2) + 3f(n+1)-f(n) = 0 for all n.

for all arithmetic progressions.

Any arithmetic progression is of the form a, a+ d, a+ 2d, ... with nth term a+ d(n-1)

The sum of n terms of that progression is a+ (a+d)+ (a+ 2d)+ ...+ a+d(n-1)= na+ d(1+ 2+...+ (n-1)). It is well known that 1+ 2+ ...+ n-1= n(n-1)/2 so that is f(n)= na+dn(n-1)/2.

Now, calculate f(n+1), f(n+2), f(n+3) and put them into the formula.
  • #5

Hello and welcome to the forum!

To answer your question, no, -3f(n+2) does not equal f(-3n-6). These are two different expressions and cannot be equated in this way.

In order to prove the given equation, we can start by expanding the left side:

f(n+3) -3f(n+2) + 3f(n+1) - f(n) = (f(n+2) + f(n+1) + f(n)) - 3(f(n+2)) + 3(f(n+1)) - f(n)

= f(n+2) + f(n+1) + f(n) - 3f(n+2) + 3f(n+1) - f(n)

= (f(n+2) - 3f(n+2)) + (f(n+1) + 3f(n+1)) + (f(n) - f(n))

= -2f(n+2) + 4f(n+1)

Now, we can use the given information that f(n) is the sum of n terms of an arithmetic progression. This means that f(n+1) is the sum of n+1 terms, and f(n+2) is the sum of n+2 terms.

Therefore, we can rewrite the equation as:

-2f(n+2) + 4f(n+1) = -2(n+2) + 4(n+1) = -2n - 4 + 4n + 4 = 2n

Since this is not equal to 0, we can conclude that the original equation is not true for all values of n.

I hope this helps! Let me know if you have any further questions.

1. What is the formula for "f(n)"?

The formula for "f(n)" is not specified in the given statement. It could represent any function of "n" such as a polynomial, exponential, or trigonometric function.

2. Can you provide an example of how to solve this equation?

Yes, for example, if "f(n)" is a linear function, the equation can be solved by substituting the function's expression into the equation and then solving for the constant term.

3. How do you prove that the equation is true for all values of "n"?

To prove that an equation is true for all values of "n", a mathematical induction method can be used. This involves showing that the equation is true for a base case, then assuming it is true for "n=k" and proving it is also true for "n=k+1". This process is repeated until it can be shown that the equation is true for all values of "n".

4. What is the purpose of this equation in the context of a forum?

The purpose of this equation in the context of a forum is likely for other members to provide solutions or proof techniques, and for discussion on different approaches and applications of the equation.

5. How can this equation be applied in real-world scenarios?

This equation can be applied in various fields such as engineering, physics, and computer science to model and solve problems. For example, it could be used to analyze the growth rate of a population or the behavior of an electrical circuit.

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