A. -x^3 and x^11 ; 3
B. 0.6 and -0.002025; 2
C. x^2,x^6,x^10.... (A sub 20) * sorry for the sub :D but I guess that's already clear? Hehe.
D. a^2/2 , a^4/4, a^6/8.... (A sub 10)
E. X+1, 2x^2 + 2x, 4x^3 + 4x^2 (A sub 6)
F. 32y^2, 16y^2, 8y^2 (A sub 8)
* sorry for the sub thingy guys. I just can't find another way to explain it. Hehe :D
Problems:
I. The cells in a culture divide every hour. Write a formula for the number of cells in the culture at the end of "n" hours. If there are 200 cells now, how many cells will there be at the end of 5 hours?
II. A piece of spoiled meat has some bacteria in it. The amount of bacteria increases four times every hour. If the amount of bacteria is 1500 on the 1st hour what's the total amount of bacteria at the end of 6 hours?
III. Three number of forms a geometric sequence and their sum is 70. If the smallest and the largest numbers are each multiplied by 4 and the middle number is multiplied by 5, the resulting products from an arithmetic sequence. Find the numbers.
I really need help. This is just hard for me :D and really big thanks to bobpursley ;) for answering the problem a while ago.
no idea what A and B are supposed to mean
You need to recognize powers of 2 and simple multiples
C: An = x^(4n-2)
D: An = x^(2n)/2^n
E: An = 2^n(x^n + x^(n-1))
F: An = 64y^2/2^n
I: 200*2^n
II: 375*4^n
III: If the numbers are x,y,z then we have
y/x = z/y
x+y+z = 70
5y-4x = 4z-5y
GP: 10,20,40
AP: 40,100,160
or the reverse works, as well
I. is 200 *2^n
if n = 5, then 200 * 32
II. 2^4 = 16 times as much every hour if it doubles every quarter hour 1500*16^6
(infinite :)
III. a r^n + a r^n+1 + a r^n+2 = 70
or
a r^2 (1 + r + r^2) = 70
4 a r^n + 5 a r^n+1 + 4 a r^n+2 is arithmetic
so
5a r^n+1 - 4a r^n = d
4a r^n+2 - 5a r^n+1 = d
or
5ar -4a = 4ar-5a
ar = -9 a
r = -9 etc
I. To find a formula for the number of cells in the culture at the end of "n" hours, we need to determine the pattern or rule that describes how the number of cells changes over time.
From the information given, we know that the cells divide every hour. This means that the number of cells doubles every hour. So, if we denote the initial number of cells as "C" and the number of hours as "n", the formula for the number of cells in the culture at the end of "n" hours can be expressed as:
Number of cells = C * 2^n
Given that there are 200 cells initially, we can substitute the values into the formula to calculate how many cells there will be at the end of 5 hours:
Number of cells = 200 * 2^5
Number of cells = 200 * 32
Number of cells = 6400
Therefore, there will be 6400 cells in the culture at the end of 5 hours.
II. In this problem, the amount of bacteria increases four times every hour. So, if the initial amount of bacteria is denoted as "B", and the number of hours is denoted as "n", the formula for the total amount of bacteria at the end of "n" hours can be expressed as:
Total amount of bacteria = B * 4^n
Given that the initial amount of bacteria is 1500, we can substitute the values into the formula to calculate the total amount of bacteria at the end of 6 hours:
Total amount of bacteria = 1500 * 4^6
Total amount of bacteria = 1500 * 4096
Total amount of bacteria = 6,144,000
Therefore, there will be 6,144,000 bacteria at the end of 6 hours.
III. In this problem, we are given that three numbers form a geometric sequence and their sum is 70. Let's represent the three numbers as "a", "ar", and "ar^2", where "a" is the first term, "r" is the common ratio, and "ar^2" is the third term.
The sum of the three numbers is given by:
a + ar + ar^2 = 70
We are also given that if the smallest and largest numbers are each multiplied by 4, and the middle number is multiplied by 5, the resulting products form an arithmetic sequence. Let's represent the new terms as "4a", "5ar", and "4ar^2".
Therefore, the new sum of the three numbers (in the arithmetic sequence) is given by:
4a + 5ar + 4ar^2
Since the sum remains the same (70), we can set up the equation:
a + ar + ar^2 = 4a + 5ar + 4ar^2
Simplifying this equation will allow us to find the values of "a" and "r" that satisfy both the geometric and arithmetic sequence conditions.
This is the general approach to solving the problem. We can take it further by substituting specific values for the sum and solving for "a" and "r", but that would require more information or constraints.
I hope this explanation helps! Let me know if you have any further questions.