BB-103

Arithmetic

t(n)=100-50n
n t(n)
0 100
1 50
2 0
3 -50

We know our initial term, 100, and we know the difference between the initial number and the number right after that, which is 100-50, which equals 50. And since this sequence is decreasing, we know that our generator is also negative. Since we are only increasing once between the independent variables, we know that our generator is -50.

f(x)=50+25x
x f(x)
0 50
1 75
2 100
3 125

We know that our initial term, which is 50, and we know that when x equals 2, f(x) is equivalent to 100. Using the elimination process in which we substitute each set of numbers in the sequence to the equation, f(x)=dx, knowing that d is our generator, x is the independent variable and f(x) is the dependent variable. So we have 50=0d and 100=2d. Thus when we subtract each, 100-50 & 2-0, we get 50 and 2. We divide 2 into 50 to get the number by which we add, or d, and we get 25 to be that number. Thus we can now finish our equation knowing the initial number and the generator. f(x)=50+25x


Geometric

t(n)=100/(2^n)
n t(n)
0 100
1 50
2 25
3 12.5

We know that when we branch from 100, the initial term, to 50, and we know that the sequence is geometric, meaning that we multiply or divide by a generator to the nth power, that we are going from a higher number, 100, to a lower number, 50, we are using division. Thus to figure out the rest of the terms in the sequence we must then divide 100 and 50 to find our generator, which is 2. Since our generator is 2 and our initial number is 100, we can make the equation stated above the table, and fill out the rest of the chart.

f(x)=50*(2^1/2)^n
x f(x)
0 50
1 50*(2^1/2)
2 100
3 100(2^1/2)

Since we know that f(x)=a+d^x, and we know that a is the initial number, the d is the generator of the sequence, and the x is the independent variable, i.e. the number of times you multiply the generator, d, due to the fact that is is a geometric sequence, we can plug in a set of numbers to help us figure out the generator of the sequence, which we do not know in this case. Since the initial number is always when x is equivalent to 0, we know that in this sequence our initial number is 50. Now instead of plugging that in, we'll use our second set of numbers, 2 and 100, which will be much easier to work with. Since 100 is equivalent to f(x) and 2 is equivalent to x, we can plug those numbers in along with our initial number, so that we get a good 100=50*d^2.