As I predicted, I was blocked on the question which asked for the explicit definition of a goemetric sequence given a pair of terms.
The problem gave us:
t3=4 and t7= (1/4)
Now that I think about it and with primary methods, the rate of change is 1/2.
The problem is, I have no idea how to get this answer in a simpler and quicker way.
On the paper I could only write that the explicit definition would look like: tn=t1*(r^n-1)
And that the # of jumps from t3 to t7 is 4.
What do I do next?
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7 comments:
Susy, I agree with you that the hardest part is getting the "r" of the sequence. However once you have it, it's very easy to solve the rest of the problem.
Assume that t(3) is term 1, so t(7) would be t(5).
If you write the explicit definition it would be
t(n)=4*(1/2)^(n-1)
To prove this, you plug t(7) to the equation, which we converted to t(5)
t(5)=4*(1/2)^(5-1)
t(5)=4*(1/2)^4
t(5)=4*(1/16)
t(5)=1/4 (which is what is given to us in the problem)
Term one is two terms before term 3, so if t(3) is t(1), then t(1) is really t(-1).
t(-1)=4*(1/2)^(-1-1)
t(-1)=4*(1/2)^(-2)
t(-1)=4*(2^2)
t(-1)=4*4
t(-1)=16
Now we can plug the real term 1 into the formula. So it will be:
t(n)=16*(1/2)^(n-1)
Check the formula...
Susy, i had the same problem. I used the formula and plugged in numbers to find the ratio, and by getting the ratio, I could've formulated the explicit definition. The problem was that I got + or - 1 which didn't make sense. Therefore, I didn't have time to fix it since I was concentrated in other questions too.
susy i also had the same problem. And thanks mac, that methos is really good. Susy but also theres another way which I did in remedials for a problem and in the quiz I blocked myself. There are four jumps and the first term given is 4. so you do t(n) = 4 x r to the power of four = the other term whic = 1/4.
4 x r to the power of four = 1/4 and then you do all the algebra you need and then find t1 and the rate.
I think Diego is on the quickest path.
4 jumps
4*r^4=1/4
Solve for r to find r = + or - 1/2 (You were right, Bee)
Both work!
Then divide 4 by each to get t2 and then divide again to get t1: 16 in both cases.
Then write the formulas; there are two different ones that work.
Nice work putting your thoughts here.
Susy if you are having trouble with this kinds of problem check mary's post. She wrote a way, I think is easier, to solve this type of problems. I was having the same difficulties as you and when mary explained the problem her way I understood alot better, You should check it out.
I was also having the problem you had, but, during the quiz I tried to solve it the way MAC did, but I'm not sure if I got the same answer she did.
i had the same problem on the quiz and because of thet i blocked i started using macs way but i didnt finish but now i can see diegos way and it is much better and easier
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