find the formula for n-th term of a sequence?

Question

Can anyone help me find the formula for the nth term of a sequence? And how do you find it?

2, -1, 1/2, -1/4, 1/8

Thanks!!!

Answer 1

This is a Geometric progression,
first term = 2
common ratio = -1/2
nth term = (2) (-1/2)^(n-1)

Answer 2

OK this is in Algebra 2.
First look at the numbers. You see

2, -1, 1/2, -1/4, 1/8

Try and find a relationship. In this case, it's multiply by -1/2.
This is a geometric sequence, because you multiply. The formula for the nth term of a geometric sequence is:

A(sub-n) = A (sub-1) * r ^(n-1)

Note: sub = subscript, ^ = to the power of
Where n is the number of the term.
A (sub-1) represents the first number in this sequence (2)
r is the common ratio (-1/2)
So the formula is:

A(sub-n) = 2(-1/2)^(n-1)

Answer 3

The sequence begins with a 2. Then each term,s absolute value is the preceding term, divided by 2, meaning the original 2 divided by 2^(n-1)

n=1 then 2^1
n=2 then 2^0
n=3 then 2^-1

n=N then 2^(2-N)

As for the alternating sign, this is done by raising (-1) to a power relted to n. When the power is even, the result is positive.

Value = [(-1)^(n+1)]*[2^(2-n)]

Answer 4

an = a1 * r^(n - 1)

a2 = 2 * r^(2 - 1)
a2 = 2 * r^1
a2 = 2r
-1 = 2r
r = (-1/2)

a3 = 2 * r^(3 - 1)
a3 = 2 * r^2
a3 = 2r^2
(1/2) = 2r^2
(1/4) = r^2
r = (1/2) or (-1/2)

a4 = 2 * r^(4 - 1)
a4 = 2 * r^(3)
a4 = 2r^3
(-1/4) = 2r^3
r^3 = (-1/8)
r^3 = (-1/2)

a5 = 2 * r^(5 - 1)
a5 = 2 * r^(4)
a5 = 2r^4
(1/8) = 2r^4
r^4 = (1/16)
r = (1/2) or (-1/2)

So as you can see, the only thing they all have in common is r = (-1/2)

the formula is

an = 2 * (-1/2)^(n - 1)

this can also be written as

an = 2*((-2)^(-1))^(n - 1)
an = 2(-2)^(-(n - 1))
an = 2(-2)^(-n + 1)

The true formula is

an = 2(-1/2)^(n - 1)

a1 = 2
r = (-1/2)

Answer 5

well it's alternating, so you know you need a (-1)^n+1

and you are dividing by two each time,

(-1)^(n) 2 * 2^(-n) (n starting at 0)

(-1)^(n+1) 2*2^(-n+1) (n starting at 1)

Answer 6

the ratio is -1/2.

this is geometric sequence, the formula is
An = a1 * r^(n-1)

a1 = first term

An = 2 * (-1/2)^(n-1)

Answer 7

n-th term is 2 * (-1/2)^(n-1) or -4* (-1/2)^n

Answer 8

r=-1/2 or
r= (-1/4)/(1/2)=-1/2

f(x)=2*[(-1/2)^(n-1)]