Question

Give a recurrence relation that describes the sequence 3, 6, 12, 24, 48, 96, 192, ....see screenshot..

Answer 1

a good first start is to see what the differences between terms are

Answer 2

Also, check if they're all multiples of each other.

Answer 3

3 6 12 24 48 96 192
3 6 12 24 48 96 <-- notice that each new term is
double the one before it

Answer 4

Which suggests that the series is given by powers of 2 multiplied by the original term.

Answer 5

if we wanted an explicit formula, yes

Answer 6

Oh yeah, recurrence relation. Just doubling will do for that.

Answer 7

\[a_n = k~a_{n-1}\]
\[a_n - k~a_{n-1}=0\]
\[r^n - k~r^{n-1}=0\]
\[r^{n-1}(r - k)=0~:~r=k\]
\[a_n=C(k)^n\]
when we know the first few terms
\[a_1=C(k)^1=M\]
\[C=\frac Mk\]
i just learneded that the other day ....

Answer 8

so if n=1 then p(n)=1?
because 1^2 =1?

Answer 9

@amistre64

Answer 10

p(1) is just the first term thats in front, it has to be defined in order to be able to actually start creating the sequence from the rule.

p(1) = 3

Answer 11

the rule can be seen from the difference levels; the second row is the number that is added to the "top left" to get to the number on the "top right": to get from 3 to 6, you have to add 3. to get from 6 to 12, you have to add 6; to get from n to n+1, you have to double n


3 6 12 24 48 96 192
3 6 12 24 48 96 <-- notice that each new term is
double the one before it

Answer 12

so it would be n^2 for the n-1?

Answer 13

No, An=2A(n-1)

Answer 14

Got it! the first answer is 3 and second one is 2! Thanks!

Answer 15

Yeah, it's An=3*2^n.