Geometric Seqences.

We learned about Geometric Sequences today. We did arithmetic sequences back in grade 11, and the difference between arithmetic and geometric sequences is that the former is linear and the latter is exponential.

To determine whether a sequence is geometric or not, we look for a common ratio.

Ex:

r = common ratio

1, 2, 4, 8, 16 . . . Is there a common ratio? YES! r? 16/8, 8/4, 4/2, 2 / 1 = 2

4, 2, 1, 1 /2, 1 /4, 1 / 8, Is there a common ratio? YES! r? 1/8 / 1/4 . . . 2/4 = 1/2

5, 15, 20, 25 Is there a common ration? NO! Arithmetic sequence.

How do we find n term in a geometric sequence?

t = { 3, 6, 12, 24 . . . }

First, we find the common ratio:

24 / 12, 12 / 6, 6 / 3 = 2

How do we solve for n when n is 10?

tn = t1*r^(n -1)
t10 = 3 * 2 ^(10 - 1)
t10 = 3 * 2 ^ (9)
t10 = 3 * 512
t10 = 1536

How about something like this?

t = { 4, 2, 1, 1 /2, 1 /4, 1 / 8 . . . }

Common ratio? (1/8) / (1/4), (1/4) / (1/2), (1/2) / 1 = 1/2

tn = t1*r^(n -1)
t10 = 4 * 1 / 2 ^(10 - 1)
t10 = 4 * 1 / 512
t10 = 4 / 512
t10 = 1 / 128


We can also use neat sequences like:

t = { 3, -6, 12, -24 . . . }

How do we figure out 10 in this one?

First, we find the common ratio:

-24 / 12, 12 / -6, -6 / 3 = -2

tn = t1*r^(n -1)
t10 = 3 * -2 ^(10 - 1)
t10 = 3 *- 2 ^ (9)
t10 = 3 * -512
t10 = -1536

No real changes here! Just a negative number. But what about an odd number for n? Looking at the sequence, we can predict that the number will be negative. Let's use 19 in this case:

tn = t1*r^(n -1)
t10 = 3 * -2 ^(19 - 1)
t10 = 3 *- 2 ^ (9)
t10 = 3 * -262144
t10 = -786432