Construct a recursive formula and prove using induction

We have the sequence 3, 7, 27, 127...
The recursive formula would then be that a(1) = 3 and a(n+1) = a(n) + 4*5^(n-1)
I need to show by induction that a(n) = 5^n + 2 for n greater than or equal to 0.
Everything goes well until I get to the induction step, where I can't at all show that the left side (5^(p+1) + 2) is equal to the right side (a(p) + 4*5^(p-1) which becomes 5^p + 2 + 4*5^(p-1)

How should I think? Where is the error? I also used the assumption in the right side while substituting a(p) with 5^p + 2

Reply 1

DFranklin

You say " I need to show by induction that a(n) = 5^n + 2 for n greater than or equal to 0."

However, you also say a(1)=3.

These two statements are incompatible.

Reply 2

Nothinghere21

Original post by DFranklin

You say " I need to show by induction that a(n) = 5^n + 2 for n greater than or equal to 0."

However, you also say a(1)=3.

These two statements are incompatible.

Then does the 3 in 3, 7, 27, 127... have an index of 1 or 0?

Reply 3

DFranklin

Original post by Nothinghere21

Then does the 3 in 3, 7, 27, 127... have an index of 1 or 0?

If you don't want to change your induction statement, you'll want to say a(0)=3, a(1)=7,...

Reply 4

Nothinghere21

Original post by DFranklin

If you don't want to change your induction statement, you'll want to say a(0)=3, a(1)=7,...

Okay so I changed the recursive definition twice now. One time to a(n+1) = a(n) + 45^n and the other time to a(n) + 20*5^(n-1) if we only take 7 as the first number. I still cannot prove that left = right for any of them

Reply 5

DFranklin

Original post by Nothinghere21

It's not clear what you're actually doing - there are 3 pieces of information:
the sequence a(n) = 3, 7, 27, ...
the relationship between a(n+1) and a(n)
the result a(n) = 5^n + 2 (that you said you wanted to prove by induction).

You need all 3 of these to be consistent (in terms of does your sequence start at n = 0 or n = 1) if you're going to get the correct result. (And if you have them consistent you should not have difficulty proving your induction).

So it's incredibly likely you're not being consistent, but from what you've posted it's hard to diagnose the exact problem.

Note also that it's currently problematic using asterisks to denote multiplication on TSR (because they are also used to denote italics) so you're probably best avoiding them.

Reply 6

mqb2766

Original post by Nothinghere21

Your recursive step consists of 3 values n+1,n,n-1 which I guess is the reason for the problem. It helps to write it out so you have the sequence
n ...... 0 .. 1 ... 2 ...3
a(n) ..3 ... 7 .. 27.. 127

So your recusive rule
a(n+1) = a(n) + 4*5^(n-1)
would give by definition a(0)=3, then a(1) corresponds to n=0 which gives 3+4/5 = 3 4/5. Clearly a problem and the fix should be fairly obvious? To check, just repeat the above and check it gives a(1) ... correctly.