Proving my conjectures using generic sequences

[mapaccanari]

Hello! I need help with making generic sequences for these three conjectures...any help? Thank you!


S1:The sequence is neither geometric nor arithmetic...(16,20,24,28,32) + (5.,10.,20.,40.,80.)= 21,30,44,68,112.


Therefore my conjecture for S1 is: When adding an arithmetic sequence with a geometric sequence the sequence sums will turn out to be neither geometric nor arithmetic. (I have to prove this using generic sequences)


S2: The sequence is arithmetic...(4,6,8,10,12)+(12,14,16,18,20)= 16,20,24,28,32


Therefore my conjecture for S2 is: When adding two arithmetic sequences together the sequence sums will turn out to be arithmetic. (I have to prove this using generic sequences)


S3: The sequence is geometric...(8,16,32,64,128)+(5,10,20,40,80)= 13,26,52,104,208)


Therefore my conjecture for S3 is: When adding two geometric sequences together the sequence sums will be geomtric and the ratio will stay 2. (I have to prove this using generic sequences)


(Note that all these sequences have the same ratio of two)

 

[stapel]

I need help with making generic sequences for these three conjectures.

A conjecture is something which is proven or dis-proven.

S1:The sequence is neither geometric nor arithmetic...(16,20,24,28,32) + (5.,10.,20.,40.,80.)= 21,30,44,68,112.

If this is a conjecture, then you need to prove that it is neither geometric nor arithmetic. If so, then simply apply the definitions.

Therefore my conjecture for S1 is: When adding an arithmetic sequence with a geometric sequence the sequence sums will turn out to be neither geometric nor arithmetic. (I have to prove this using generic sequences)

Are you saying that you have been given this "statement", and that your task is to attempt to generalize from this particular statement to some general formulation (the "conjecture"), which you then have to prove?

I have the same confusion for the other two "conjectures".

 

[mapaccanari]

All I have to do is using these conjectures, make genetic sequences proving them.

 

[Ishuda]

IMO, Your conjectures are (almost) correct. Your 'and the ratio will stay 2' has to be generalized. Thus
S1: Given an arithmetic and geometric sequence, their sum is neither arithmetic nor geometric.
S2: Given two arithmetic sequences, their sum is arithmetic.
S3: Given two geometric sequences with the same common ratio, their sum is geometric with that ratio.

To prove these, write a (two) general expression(s) for an arithmetic sequence and/or a geometric sequence and add these two expressions. S1 seems to me to be the hardest to prove since it 'negative proof', i.e. prove something doesn't exist. I'll do S2 for you as as an example. Given two arithmetic sequences x_n and y_n
x_n = x₀ + a * n
y_n = y₀ + b * n
their sum z_n is given by
z_n = (x₀ + y₀) + (a + b) * n
with initial term
z₀ = x₀ + y₀
and common difference
c = a + b

Is that enough for now or are you still stuck?

 

[HallsofIvy]

"Generic sequences" does NOT mean choosing specific sequences as examples. Examples can never prove a general theorem. "Generic sequences" are, as Ishuda said, "a, a+ d, a+ 2d, ..." or "{a+ nd} for n a non-negative integer" for an arithmetic sequence and "b, br, br^2, br^3, ..." or "{br^n}" for n a non-negative integer.

The sum of those sequences is "{br^n+ a+ nd}". You want to show that there cannot be numbers, c, e, such that br^n+ a+ nd= c+ ne for all n (so the sum is not arithmetic) and the cannot be numbers, p, q, such that br^n+ a+ nd= pq^n for all n (so the sum is not geometric).