What is the growth rate of these sequences?

[Ian_Brooks]

Homework Statement

Homework Equations

The mark scheme is 2, 2, 3, 3

For a) ii), it's a sequence of integers up to the floor of n²/2

The Attempt at a Solution

We haven't done growth rates of sequences, only of equations. a) i) looks like the growth rate would be 5ⁿ, but I don't know how to show the calculation of that

a) ii) difference between the differences is one, so there's an n² in the term equation. That's all I can figure out.. it seems a bit trivial but would I plug in n² to the final term, getting a quartic growth? How would I explain this properly to answer the question?

b) i) f(n) > 0, and I get to the point of showing that the equations for n=3 and n=4 etc have x_n gives a sum of f and previous x_n values, making the result have a majority of positives ( f(n) >0 ) but I'm having trouble showing that this is true for all n, mathematically. I can't just say "as n continues, the value for x > 0 since it's a sum of f(n) values, which are all positive" after only writing the value of x₃ and x₄. Also, I don't think this is true. I need to be able to show that this is a positively growing function. I don't know if the sequence of numbers in f(n) is increasing or decreasing either

b) ii) The function part is replaced by the root and the n5ⁿ, and again, we haven't done growth rates of sequences in class, only of equations. Do I turn this into a explicit form of the sequence? We haven't worked with fractional powers, so I don't know how to approach this method either. Our classes are really unfair: metaphorically; they expect us to write essays only after knowing the alphabet, no grammar is taught.

 

[willem2]

a1) If you divid e a_n by 5^(n-1) what is the growth rate of the quotient?
a2) it's not difficult to find an exact formula for 1+2+3+...+n. you can substitute floor(x/10) in that.

b1) try to prove a much stronger statement about the growth rate of x_n by induction