Duane Said:

Label the following sequences as geometric or arithmetic?

We Answered:

2, -4, 8, -16 is geometric, r= -2
1, 2, 3, 5, 8 is neither.

Eileen Said:

What is arithmetic, geometric and harmonic means of the three roots of qubic polynomial x³ - ex² - x + π?

We Answered:

if the roots are a, b, c then
P(x) = x³ - (a+b+c)x² + (ab+ac+bc)x - abc

So a+b+c = e
abc = -π
ab+ac+bc = -1
1/a + 1/b + 1/c = (ab+ac+bc)/abc = 1/π

arithmetic mean = (a+b+c)/3 = e/3
geometric mean = ³√abc = -³√π
harmonic mean = 3 / (1/a + 1/b + 1/c) = 3π

Ricky Said:

Arithmetic and geometric sequence integrated problem?

We Answered:

The 3 numbers are x- 4, x and x + 4

Then x - 2, x + 3 and x + 9 are a geometric sequence

then (x + 3) / (x - 2) = (x + 9) / (x + 3)

(x + 3)² = (x + 9)(x - 2)

x² + 6x + 9 = x² + 7x - 18

x = 27

the 3 numbers are 23, 27 and 31

Nina Said:

how do i know if its geometric or arithmetic?

We Answered:

>> the geometric series

Un = ar^(n-1) , ratio = r = Un / U(n-1)

Sn = a((r^n) - 1) / (r - 1) if r > 1

Sn = a(1 - (r^n)) / (1 - r) if r < 1

>> the arithmatic series

Un = a + (n - 1)d , difference = d = Un - U(n-1)

Sn = ½ n(a + Un)

>> the case

Un = 24(- ½)^n

let's check this out :

U1 = 24(- ½) = - 12

U2 = 24(- ½)^2 = - 6

U3 = 24(- ½)^3 = - 3

if the form on above is the aritmatic series so :

U2 - U1 = - 6 - (- 12) = 6

U3 - U2 = - 3 - (- 6) = 3

then

U2 - U1 ≠ U3 - U2

-- FALSE --

if the form on above is the geometric series so :

U2/U1 = - 6 / - 12 = ½

U3/U2 = - 3 / - 6 = ½

then

U2/U1 = U3/U2

-- CORRECT --

so Un = 24(- ½)^n ---> it's must be the geometric form

and then

Sn = a(1 - (r^n)) / (1 - r)

Sn = - 12(1 - (½)^n) / (1 - ½)

Sn = - 24(1 - (½)^n)

then

n = 5

S5 = - 24(1 - (½)^5)

S5 = - 93/4

Jorge Said:

How can I tell if it's a geometric or arithmetic sequence?

We Answered:

An arithmetic sequence is one where the difference between terms is a constant. Hence, the n+1 term is defined as the nth term plus the constant. In your example, the constant is -2, and therefore you would obtain the nth term by adding -2 to the n-1 term. There is also a zeroth term asub0, which is the other piece of information needed to define the sequence.

Geometric sequences have a constant factor by which the previous term is multiplied to obtain the next one. Again, in your example the constant is 1/2. As in arithmetic sequences, you also need to know the first term to define the sequence.

For example, if asub0 = 1, and the constant is 1/2:

The arithmetic sequence would be:

1, 3/2, 2, 5/2, 3, ...

and the geometric sequence would be:

1, 1/2, 1/4, 1/8, 1/16, ...

Suzanne Said:

Can you solve this arithmetic and geometric sequence?

We Answered:

The answer is:

6 9 12 16

6 9 12 is an arithmetic sequence based on 3

9 12 16 is a geometric sequence with a multiplier of 4/3

Ronnie Said:

how can we make a model on arithmetic progression or on geometric progression?

We Answered:

Get some square dowels with side somewhere in the 1 to 2 cm or 1/2 to 1 inch range.
Something like this: http://www.hobbylinc.com/gr/mid/mid12.jp…

Then for the arithmetic progression, cut lengths of 1, 2, 3, 4, 5, 6, etc .. times the side length.
(or any other arithmetic progression you like: 1, 1.5, 2, 2.5, 3, 3.5 or 2 4 6 8 10)

Then place them next to each other and show that they form a 'staircase':
(I'll use O to be one unit of length)
......O
.....OO
...OOO
OOOO

For the geometric progression make the lengths ... a geometric progression
such as
ratio 1.5: 1, 1.5, 2.25, 3.375, 5.0625, 7.6, 11.4 ...
which might require pretty fine measuring if the unit is small
or
ratio 2: 1, 2, 4, 8, 16, 32 ... which gets pretty long pretty fast.
Then your model will look like this:
......... ..O
......... ..O
......... ..O
......... ..O
........ .OO
........ .OO
....... OOO
......OOOO
since geometric progressions grow much faster than arithmetic.

To best show the difference, make the first two elements of each progression the same:
e.g., 1, 2, 3, 4, 5 and 1 , 2, 4, 8, 16
or 1, 1.5, 2, 2.5, 3 and 1, 1.5, 2.25, 3.375, 5.0625

Another way to show the difference would be to build the model of the
geometric progression and then put marks on each dowel showing where
the corresponding value of the arithmetic progression would be.

So make lengths of 1,2,4,8,16
and put marks are distance 1,2,3,4,5 on each of those.

Or make a two-layer model with the arithmetic progression on top of the geometric progression.