The POD for today was:
S is a geometric sequence. T3= -7 T6= -2401
Find t10
What we have to do, in order to solve the problem is to find the rate. To do so, we plug in the equation t3 as t1.
-2401= -7*r^3
We use three as the exponent because there are three jumps from t3 to t6.
Solving for r, me stay with this:
√(r^3)= √(343) - Then, we cancel the exponent on r, cubing the root of 343. The cube root of 343 is 7.
After finding the rate, we can find t10. Knowing that we used t3 as t1, we can use t8 as t10 to find the answer to the problem.
t8= -7 * (7)^(8-1)
t8= -7 * (7)^7
t8= -5, 764,801, which is t10
After finishing the POD, we started with infinite sequences.
Infinite Sequence: A sequence that goes on forever. They may or may not have finite limits.
Another way to think of a limit, is to think of it as a boundary. Some can be reached, and some can't be reached, but you can't exceed it.
The bouncing ball example
The bouncing ball example had a limit. 100,50,25,...
As n gets lower, what is the value that tn is approaching?
" The limit of tn as n approaches infinity is 0"
Limtn = 0
n-->OO
Limtn= tells you to find the limit of a sequence.
Example 1: The sequence tn=n
1,2,3,4,5,... the limit is Limtn= 0
n-->OO
If a limit exists, it has to be a real number.
The Limit Does Not Exists. (DNE)
Limtn= 0
n-->OO
Example 2: -1, -4. -9, -16
What is the limit? Answer: DNE..it will keep going on to negative infinity. It also can be:
Limtn= - OO
n-->OO
Example 3: There are grains of salt in an infitnite chess board.
2,4,8...
What is the Limit?
Limtn= -OO
n--> OO
Example 4:
1/2, 3/4, 7/8, 15/16,...
What is the limit?
As you can see, the number is approaching to 1, but it never gets to 1. So the answer is:
Limtn: 1
n-->OO
Example 5: Find Limit, if tn =(.99)^n
The limit is 0. Each time you get .99 of the number so it decreases, but never reaches 0.
The safest way to do it:
-Plug large values of n
-List terms
-Repeated multiplication of a number less than 1.
-99%
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