Can 7 divide 3^(2n+1) + 2^(n+2) in induction for number theory?

In summary, induction in number theory is a method of proof used to show that a statement holds true for all natural numbers. It involves three steps: the base case, the inductive hypothesis, and the inductive step. There are two types of induction: weak and strong. Induction can only be used to prove statements that are true for all natural numbers and is closely related to recursion, which is used to define functions or sequences.
  • #1
mcooper
29
0

Homework Statement



Show 7 divides 3^(2n+1) + 2^(n+2)

The Attempt at a Solution



Have proved base case K=1 and for the case k+1 I have got ot the point of trying to show 7 divides 9.3^(2k+1) + 2.2^(k+2).

Any pointers would be much appreciated. Thanks in advance
 
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  • #2
Well, you don't seem to have even stated the inductive hypothesis yet, let alone tried doing something with it.
 
  • #3
The inductive hypothesis is that 7 divides 3^(2k+1) + 2^(k+2)
 

FAQ: Can 7 divide 3^(2n+1) + 2^(n+2) in induction for number theory?

What is induction in number theory?

Induction in number theory is a method of mathematical proof used to show that a statement holds true for all natural numbers. It is based on the principle that if a statement is true for the first natural number, and if it is also true for the next natural number after that, then it must be true for all natural numbers.

How is induction used in number theory?

Induction is used in number theory to prove statements about natural numbers. It involves three steps: the base case, where the statement is shown to be true for the first natural number; the inductive hypothesis, where it is assumed that the statement is true for a particular natural number; and the inductive step, where the statement is shown to be true for the next natural number. By repeating this process, the statement is proven to be true for all natural numbers.

What is the difference between weak and strong induction?

In weak induction, the inductive hypothesis only assumes that the statement is true for a particular natural number. In strong induction, the inductive hypothesis assumes that the statement is true for all smaller natural numbers as well as the particular number. Strong induction is a more powerful form of induction, but both methods can be used to prove statements about natural numbers.

Can induction be used to prove all statements in number theory?

No, induction can only be used to prove statements that are true for all natural numbers. It cannot be used to prove statements that are only true for a finite set of natural numbers or for infinitely many natural numbers. In these cases, other methods of proof, such as contradiction or direct proof, may be more appropriate.

How is induction related to recursion?

Induction and recursion are closely related concepts. Induction is used to prove statements about natural numbers, while recursion is a method of defining functions or sequences in terms of themselves. Both methods rely on the principle of using previous steps to prove the next step, with induction used for proofs and recursion used for defining functions or sequences.

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