A rare bacterial culture is being grown in a lab. As the days progress, the cells multiply and grow.
After 1 day, there is only 1 cell.
After 2 days, there are 5 cells.
After 3 days, there are 10 cells.
After 4 days, there are 16 cells.
Which recursive equation represents the pattern?

Question

A rare bacterial culture is being grown in a lab. As the days progress, the cells multiply and grow.
After 1 day, there is only 1 cell.
After 2 days, there are 5 cells.
After 3 days, there are 10 cells.
After 4 days, there are 16 cells.
Which recursive equation represents the pattern?

ranga · Answer

The differences form an arithmetic sequence.

anonymous · Answer

But the differences are different

ranga · Answer

The differences form the sequence: 4, 5, 6, ......
This is an AP (arithmetic progression).
The nth term is given by the formula: a + (n-1)d = 4 + (n-1)*1 = n + 3.

ranga · Answer

The differences form an AP with the nth difference being (n+3)  
Put n = 1, 2, 3, 4, etc.  and you will get the sequence of differences.

On the left hand side of the equation we have a2 - a1 as the first difference.
a3 - a2 as the second difference.  
So the nth difference would be:$$\Large a _{n+1} - a _{n} = n + 3$$

anonymous · Answer

I'm not understanding any of this tbh

ranga · Answer

a(subscript n) is the number of bacteria on the nth day
a(subscript n+1) is the number of bacteria on the (n+1)th day

We start with a1 = 1
And we have found a recursive formula to find a2, a3, a4, etc.

Put n = 1, 2, 3, 4 in the equation I have above and you will see what it represents.

Put n = 1
a2 - a1= 1 + 3 = 4
a2 = 4 + a1 = 4 + 1 = 5

Put n =2
a3 - a2 = 2 + 3 = 5
a3 = a2 + 5 = 5 + 5 = 10

anonymous · Answer

but these are set up differently a. an = an - 1 + 2n
b. an = an - 1 + n2
c. an = an - 1 + (n + 2)
d. an = an - 1 + 2(n + 1)

ranga · Answer

In my case, n runs from 1 to infinity.
In their case, it runs from 2 to infinity.

All you have to do is replace n by n-1 in my formula everywhere in my formula to get the answer.  They are the same recursive equations but they differ in where n starts.

anonymous · Answer

would it be  an = ((an - 1)) + (n + 2)?

ranga · Answer

$$\Large a _{n+1} - a _{n} = n + 3 \quad \text{ for n = 1, 2, 3, ...}$$Replace n by (n-1)$$\Large a _{n-1+1} - a _{n-1} = (n-1) + 3$$$$\Large a _{n} - a _{n-1} = n + 2 \quad \text{ for n = 2, 3, 4, ...}$$

ranga · Answer

Yes.

ranga · Answer

If I had seen the answer choices along with the problem initially I would have derived the formula with a(n) and a(n-1) from the start.  But it is no big deal.  Both represent the same but they just differ in the starting value of n.

anonymous · Answer

Okay thank you!

ranga · Answer

alright. yw.