can you please help me

Question

Enter a recursive rule and an explicit rule for the geometric sequence, then find the next three terms.

3,072, 1,536, 768,...

The recursive rule is f (1)= _______ and f (n) = f( ) - ( )

The explicit rule is f (n) = _______

The next three terms are _____, _______, and ________

Answer 1

Let's put the terms of the sequence in a little table to help see things:

It appears that we go to the next term by dividing by 2.

So, to get to the 4th term, we take the value of 3rd term and divide by 2. To write that in math language, if we define n = 4 (the term number is 4), then the value, f(n), is the value of term number 3, aka f(3) or aka f(4-1) and divide it by 2. So we could say f(n) = f(n-1)/2. That is the recursive rule. I see your template has a subtraction sign in it. But this is a geometric sequence and we get to the next term by multiplication. Since dividing by 2 is the same as multiplication by 1/2, we can call it multiplication to get to the next term. So we can also write the recursive rule as f(n) = 1/2f(n-1). Please be aware though, that the recursive rule ONLY works if you have a place to start so you MUST specify a term value and number. It's convenient to specify f(1) so we need to say f(1) = 3072 in addition to f(n) = 1/2f(n-1).

To write the explicit rule, we need to realize that this is an exponential type problem. That means it will fit th form of y = ab^x or, in this case f(n) = abⁿ where "a" and "b" need to be determined for every unique sequence. "a" is the number when the sequence term is zero (although, that isn't a real term, we can use it to write the explicit rule). That means we need to go backwards from f(1). To do that, we can multiply the first term by 2. 3072(2) = 6144, so a = 6144. "b" is the multiplier, so b = 1/2. That makes the explicit equation: f(n) = 6144(1/2)ⁿ

There are methods that use the first term instead of the zero term to write the explicit equation. To do that, we use the first term, 3072, but then we need to adjust the exponent to accommodate the fact that we did not start at the zero term by using "n - 1" instead of "n". So, another way to write the explicit equation is:

f(n) = 3072*(1/2)^n-1

Both f(n) = 6144(1/2)ⁿ and f(n) = 3072*(1/2)^n-1 produce the same results and are correct. Try them out for different values of "n" to prove it to yourself.

The next 3 terms are, I think, most easily found by using the recursive equation, just divide term 3 by 2 and repeat. So term 4 is 768/2 = 384 etc.