Question

Tell whether the sequence 1/3,0,1,-2… is arithmetic, geometric, or neither.

Find the next three terms of the sequence.

Answer 1

Choices:
A. \[neither; 7, -20, 61\]
B. \[geometric; 7, -20, 61\]
C. \[arithmetic; -\frac{ 1 }{ 3 }, 1 \frac{ 1 }{ 3 }, 3\]
D. \[geometric; -3\frac{ 1 }{ 3 }, -5\frac{ 5 }{ 9 }, -9\frac{ 7 }{ 27 }\]

Answer 2

What are the characteristics of a geometric and arithmetic progression ?

Answer 3

geometric is when you multiply and arithmetic is when you add.

Answer 4

In geometric the ratio of successive terms is equal.
In arithmetic, difference of successive terms is equal.
Does your given sequence follow any of the two above ?

Answer 5

as we discussed yesterday, this is neither arithmetic (different differences between first two terms and second two terms) nor geometric (different ratios between first two terms and second two terms)

On to the pattern:

\[\frac{1}{3},~0,~1,-2\]

Let's look at the differences:

\[0-\frac{1}{3} = -\frac{1}{3}, 1-0 = 0, -2-1 = -3\]\[-\frac{1}{3},0,-3\]Anything interesting occur to you looking at that?

Answer 6

Um....I don't know..

Answer 7

Ok..I got it now!:) I see what you did!:)

Answer 8

But....that isn't in any of my answers...

Answer 9

the "differences" are actually
-1/3, 1 , -3

so, next terms would be,
9, -27, 81

but these are the differences!!

so, next term = -2+9=...
then, add -27
then retrice81

Answer 10

then add 81 ***

Answer 11

But you know my original problem is: 1/3, 0, 1,- 2? Will that make any difference in what you have explained above?

Answer 12

Ok. So it would be 7, -20 and 61. How would I know if it's geometric or neither?

Answer 13

@whpalmer4 Can you finish helping me?

Answer 14

what if we multiply each term by some number and add some number to it?

Answer 15

Let's assume that the solution is like that until we prove otherwise. Can you figure out what the number is that we are adding? Hint: {..., 0, 1, ...}

Answer 16

you still have doubts in this ?

Answer 17

multiply each number by -3, then add 1...


1/3 * -3 = -1, + 1 = 0
0*-3 = 0, + 1 = 0
1*-3 = -3, + 1 = -2
-2*-3 = 6, + 1 = 7
7*-3 = -21, + 1 = -20
-20 * -3 = 60, + 1 = 61

Answer 18

sorry, second line should be

0*-3 = 0, + 1 = 1

Answer 19

We would write that with a recursive definition of \[a_{n+1} = -3a_n+ 1, ~a_1 = \frac{1}{3} \]