Learning a technique to figuring out the Explicit Formula

[trigger352]

I have a lot of trouble trying to figure out the explcit forumla of a series of numbers.

I can see the pattern in a recursion forumla, however.

$$2, -4, 8, -16, ...$$ Is a multiplication of $$-2$$ to the term before it. Which is cake to write in a recursion forumla. But what about an Explicit Formula?

What techniques and ideas do you look for first? Are there any clues?

 

[dextercioby]

That is a SEQUENCE...U need to find the expression for the general term...

It's not difficult.

$$a_{1}=2$$
$$a_{n+1}=-2a_{n},\forall n\geq 1$$

Now find a_{n} as a function of "n"...

Daniel.

 

[trigger352]

That is a SEQUENCE...U need to find the expression for the general term...

It's not difficult.

$$a_{1}=2$$
$$a_{n+1}=-2a_{n},\forall n\geq 1$$

Now find $$a_{n}$$ as a function of "n"...

Daniel.

Whoa. Wait, what?

Can you breakdown this forumla for me:
$$a_{n+1}=-2a_{n},\forall n\geq 1$$
?

 

[trigger352]

That is a SEQUENCE...U need to find the expression for the general term...

It's not difficult.

$$a_{1}=2$$
$$a_{n+1}=-2a_{n},\forall n\geq 1$$

Now find $$a_{n}$$ as a function of "n"...

Daniel.

Whoa. Wait, What?

Can you breakdown this formula for me:
$$a_{n+1}=-2a_{n},\forall n\geq 1$$
?

 

[Data]

The $n+1$st term is the $n$th term multiplied by $-2$. That's all that equation says. You then need to solve the equation for $a_{n}$ in terms of $n$. It is what is called a "first order linear homogeneous constant-coefficient difference equation."

Here is a hint as to how to solve it: Guess the solution $a_n = Ak^n$ for some constants $k$ and $A$, and try to determined what $k$ is by substitution into the equation. Then solve for $A$ by using the initial condition $a_1 = 2$

 

[dextercioby]

That's called "reccurence relation".It defines a sequence of numbers...

There's no guessing here.It's a simple geometric progression with the ratio "-2".

Daniel.