Find the pattern and write formula
3, 6, 12, 24, 48, 96
I understand that the number doubles each time, but I am quite confused as to how to figure out the formula for this.
Another problem that I'm having trouble with:
32, -16, 8, -4, 2 ,-1
Thanks a lot
well you found the pattern yourself in the first two, you realized one was being doubled and the other was being halved. so i tried to write the terms out in terms of what was happening. sorry you didn't get it. let's try this one step by step.
1, 3/2, 5/4, 7/8, 9/16, 11/32....
ok, i notice that the denominator keep increasing by a factor of 2, we can see this if we focus only on the denominator, say like this, call all the tops 1 for the time being
1*(1/2) = 1/2
1/2 * 1/2 = 1/4
1/4 * 1/2 = 1/8
notice that for the denominators, you get every new one by multiplying the old one by 2, that's the pattern for the denominators. so i expect my common ratio to look something like (a/2)^n
now focus on the numerators, how are they changing?
we have 1 then 3 then 5 then 7 then 9...notice a pattern? these are all consecutive odd numbers. odd numbers are given by the formula, 2n+1, so we expect our formula to have the form 2n+1(1/2), but now what are the powers? we want to use n=1,2,3,4,5...
so for the first i want 1*1 then 1*3/2 then 1*5/4
so if we say (2n+1)(1/2)^n, for n=1,2,3,4,5... we get the sequence
3/2, 5/4, 7/8,... almost our sequence, but not completely, we want the term before 3/2 to be included, so we make all the n's into n-1's.
so we get a_n = (2(n - 1)+1)(1/2)^(n-1), for n = 1,2,3,4,5...
or a little more nicey, a_n = (2n-1)(1/2)^(n-1) for n=1,2,3,4,5...
or maybe even (2n - 1)/2^(n-1)
the sequence is: 1, 3/2, 5/4,...