To extend the geometric part a bit: even if we allow a constant term, k levels of composition leads to a tower of order k+1 (still with a constant factor and a constant term, of course). To make it specific, let's define: x_(k,n) = c_k + a_k*r_k^n, k>=1 y_(1,n) = x_(1,n) y_(k,n) = x_(k,y_(k-1,n)), k>=2 Then, we can calculate: y_(2,n) = c_2 + (a_2*r_2^c_1)*(r_2^a_1)^r_1^n y_(3,n) = c_3 + (a_3*r_3^c_2)*(r_3^(a_2*r_2^c_1))^(r_2^a_1)^r_1^n y_(4,n) = c_4 + (a_4*r_4^c_3)*(r^4(a_3*r_3^c_2))^(r_3^(a_2*r_2^c_1))^(r_2^a_1)^r_1^n ... In general, y_(k,n) = c_k + A_k*R_k^R_(k-1)^R_(k-2)^...^R_2^R_1^n, where the new symbols are given by: A_j = a_j*r_j^c_(j-1) R_j = r_j^A_(j-1) using the convention that c_j = 0 and a_j = r_j = 1 for j
The reason it is done this way is so that a_1 = a and g_1 = b. The way you defined it a_0 = a and g_0 = b, which is also fairly reasonable, and in my opinion zero-indexed sequences are usually cleaner, but starting at 1 is a common convention.
Everything you've talked about seems entirely generalisable to composing linear and exponential functions. I think it helps the intuition, for me at least.
I paused in the beginning thinking that the problem would be to determine when a sub g sub n = g sub a sub n. This problem has the not very exciting solution a_n = g_n = b, a constant. Which is indeed an arithmetic series (difference 0) as well as a geometric series (common ratio 1).
This video simplifies subsequences of arithmetic and geometric sequences with clear explanations and examples. It’s a great resource for understanding patterns in sequences. Tools like SolutionInn’s AI features have helped me a lot with similar problems worth exploring to boost your learning!
9:42 best term can be suggested here is "tetration sequence"
To extend the geometric part a bit: even if we allow a constant term, k levels of composition leads to a tower of order k+1 (still with a constant factor and a constant term, of course). To make it specific, let's define:
x_(k,n) = c_k + a_k*r_k^n, k>=1
y_(1,n) = x_(1,n)
y_(k,n) = x_(k,y_(k-1,n)), k>=2
Then, we can calculate:
y_(2,n) = c_2 + (a_2*r_2^c_1)*(r_2^a_1)^r_1^n
y_(3,n) = c_3 + (a_3*r_3^c_2)*(r_3^(a_2*r_2^c_1))^(r_2^a_1)^r_1^n
y_(4,n) = c_4 + (a_4*r_4^c_3)*(r^4(a_3*r_3^c_2))^(r_3^(a_2*r_2^c_1))^(r_2^a_1)^r_1^n
...
In general, y_(k,n) = c_k + A_k*R_k^R_(k-1)^R_(k-2)^...^R_2^R_1^n, where the new symbols are given by:
A_j = a_j*r_j^c_(j-1)
R_j = r_j^A_(j-1)
using the convention that c_j = 0 and a_j = r_j = 1 for j
Proof by induction in j:
A_0 = a_0*r_0^c_(-1) = 1*1^0 = 1
A_1 = a_1*r_1^c_0 = a_1*1^0 = a_1
R_1 = r_1^A_0 = r_1
R_2 = r_2^A_1 = r_2^a_1
Base case:
y_(1,n) = c_1 + a_1*r_1^n = c_1 + A_1*R_1^n
Induction step:
y_(k+1,n) = x_(k+1,y_(k,n)) = c_(k+1) + a_(k+1)*r_(k+1)^y_(k,n)
= c_(k+1) + a_(k+1)*r_(k+1)^(c_k + A_k*R_k^R_(k-1)^R_(k-2)^...^R_2^R_1^n)
= c_(k+1) + (a_(k+1)*r_(k+1)^c_k)*(r_(k+1)^A_k)^R_k^R_(k-1)^R_(k-2)^...^R_2^R_1^n
= c_(k+1) + A_(k+1)*R_(k+1)^R_k^R_(k-1)^R_(k-2)^...^R_2^R_1^n
Wouldn't most calculations have been much simpler had you let n start at 0 and defined a_n = a+dn and b_n = br^n? Am I missing something?
Classic mathematicians not understanding 0 indexing being easier.
The reason it is done this way is so that a_1 = a and g_1 = b. The way you defined it a_0 = a and g_0 = b, which is also fairly reasonable, and in my opinion zero-indexed sequences are usually cleaner, but starting at 1 is a common convention.
programmers vs. mathematicians
Everything you've talked about seems entirely generalisable to composing linear and exponential functions. I think it helps the intuition, for me at least.
Is there any reason behind this video
Or it is just for fun and an interesting thing
Just a fun exercise in working with subsequences, and also working with the nth term formula for arithmetic and geometric sequences.
I paused in the beginning thinking that the problem would be to determine when a sub g sub n = g sub a sub n. This problem has the not very exciting solution a_n = g_n = b, a constant. Which is indeed an arithmetic series (difference 0) as well as a geometric series (common ratio 1).
This video simplifies subsequences of arithmetic and geometric sequences with clear explanations and examples. It’s a great resource for understanding patterns in sequences. Tools like SolutionInn’s AI features have helped me a lot with similar problems worth exploring to boost your learning!