Just for fun - number sequence

The 1st term in a geometric(?) number sequence is 2. The 5th term is 14.

what is the formula for the sequence?

edit: hmm.. maybe you need one more term. OK, the 57th is 170.

Perhaps you could refresh my memory and illustrate what the form of a geometric series looks like?

Is it something like:

X_i = A*X³_(i-1) + B*X²_(i-1) + C*X_(i-1) + D


???
-Lou

I'm not sure this is a geometric sequency, that's why I put it in brackets.

I think a geometric sequence is one where you multiply the number by a number which is constant. Then it can add or subtract number afterwards.
eg
U_n = 2n+1
would give
3, 5, 7, 9 ...

thats the aritmetic series
the geometric series goes exponential:

a+aq+aq^2+..+aq^n = a(1-q^n)/(1-q), q/=1

a=2
2q^5=15 gives q=7.5^0.2~1.496

you don't really need 57'th term, but lets see if you made this up..

2*7.5^(0.2*57)~18911605435 obviously not 170

lets give the aritmetic one a shot (sorry i didn't notice it was you who posted that)
the n'th number should be a+nd where a is the first

2+5d=14 <=> d= 12/5 = 2.4
2+57d=170 <=> d=168/57 = 2.9

doesn't work out...

X_n = 3n - 1

So,
N=1 => X=2
N=5 => X=14
N=57 => X=170 (3*57 - 1 =171 - 1 = 170)

Its just a linear equation.
-Lou

Your right NotLKH, how did you get it?

14 - 2 = 12 (Difference between 5th term and 1st term)

12 / 4 = 3 (Divide by 4 as there is a difference of 4 terms between the first and fifth)

=> coefficient of n is 3

So far Un = 3n

Using that for the first term gives 3.

2 - 3 = - 1

=> constant is - 1

Therefore:

Un = 3n - 1

(This method only works for linear equations though)

And on the TI-83 you could've used a linear regression, although Celest's way is better ;)

Just a point.

These numbers have a constant difference between terms (in this case 3) That makes it an Arithmetic sequence.

A Gemetric sequence is one with a common ration between terms. (as kedaman said)

Now, if you're given an Nth value and a Kth value (i.e. two indexed values) in either you can find the sequence (unless it is a geometric sequence, and (N-K) is even, then there are two possibilities)

I.e. for Geometric series:
5th term: 14
1st term: 1
=> Nth term = 14^(n-1)

the aritmetic series first element would be -1, not 2..

T_n = 3n - 1

The first term, T₁ is 2, the 0^th term, T₀, would be -1.

there is no 0th term, in math or in english. You start counting from 1