Edexcel A Level Further Maths: Core Pure:复习笔记9.1.2 Common Cases of Proof by Induction

What are the steps for proof by induction with sequences?

• STEP 1: The basic step
  • Show the result is true for the base case
    • If the recursive relation formula for the next term involves the previous two terms then you need to show the position-to-term formula works the first two given terms which will be given as part of the definition of the sequence
  • This is normally n = 1 or 0 but it could be any integer
    • For example: To prove  is the position-to-term formula for the recursive sequence :
    • 
• STEP 2: The assumption step
  • Assume the result is true for n = k for some integer k
    • For example: Assume  is true
  • There is nothing to do for this step apart from writing down the assumption
• STEP 3: The inductive step
  • Using the assumption show the result is true for n = k + 1
  • It can be helpful to write down what you want to show
  • The assumption from STEP 2 will be needed at some point
    • For example:  Want to show
  • The trick to this step is to use the recursive relation formula:
    • 

• STEP 4: The conclusion step
  • State the result is true
  • Explain in words why the result is true
  • It must include:
    • If true for n = k then it is true for n = k + 1
    • Since true for n = 1 the statement is true for all n ∈ ℤ, n ≥ 1  by mathematical induction
  • The sentence will be the same for each proof just change the base case from n = 1 if necessary

What are the steps for proof by induction with series?

• STEP 1: The basic step
  • Show the result is true for the base case
  • This is normally n = 1 or 0 but it could be any integer
    • For example: To prove  is true for all integers n ≥ 1 you would first need to show it is true for n = 1:
    • 
• STEP 2: The assumption step
  • Assume the result is true for n = k for some integer k
    • For example: Assume  is true
  • There is nothing to do for this step apart from writing down the assumption
• STEP 3: The inductive step
  • Using the assumption show the result is true for n = k + 1
  • It can be helpful to simplify LHS & RHS separately and show they are identical
  • The assumption from STEP 2 will be needed at some point
    • For example:  and
  • The trick to this step is to use:

• STEP 4: The conclusion step
  • State the result is true
  • Explain in words why the result is true
  • It must include:
    • If true for n = k then it is true for n = k + 1
    • Since true for n = 1 the statement is true for all n ∈ ℤ, n ≥ 1  by mathematical induction
  • The sentence will be the same for each proof just change the base case from n = 1 if necessary

What are the steps for proof by induction with series?

• STEP 1: The basic step
  • Show the result is true for the base case
  • This is normally n = 1 or 0 but it could be any integer
    • For example: To prove  is divisible by 3 for all integers n ≥ 1 you would first need to show it is true for n = 1:
    • 
• STEP 2: The assumption step
  • Assume the result is true for n = k for some integer k
    • For example: Assume  is divisible by 3
  • There is nothing to do for this step apart from writing down the assumption
  • The trick for this step is to write the expression as a multiple of the number
    • For example:  for some integer p
• STEP 3: The inductive step
  • Using the assumption show the result is true for n = k + 1
  • The assumption from STEP 2 will be needed at some point
    • For example: Show that  is divisible by 3
  • The trick to this step is to use:

• • • For example:
  • Another trick is to show that  is divisible by the number
  • Be careful with numbers like 6 as for these you can instead show it is divisible by both 3 and 2

• STEP 4: The conclusion step
  • State the result is true
  • Explain in words why the result is true
  • It must include:
    • If true for n = k then it is true for n = k + 1
    • Since true for n = 1 the statement is true for all n ∈ ℤ, n ≥ 1  by mathematical induction
  • The sentence will be the same for each proof just change the base case from n = 1 if necessary

What are the steps for proof by induction with matrices?

• STEP 1: The basic step
  • Show the result is true for the base case
  • This is normally n = 1 but it could be any integer
    • For example: To prove  where  for :
    • 
• STEP 2: The assumption step
  • Assume the result is true for n = k for some integer k
    • For example: Assume  is true
  • There is nothing to do for this step apart from writing down the assumption
• STEP 3: The inductive step
  • Using the assumption show the result is true for n = k + 1
  • It can be helpful to write down what you want to show
  • The assumption from STEP 2 will be needed at some point
    • For example:  Want to show
  • The trick to this step is to use write:
    • 
    • The entries in the matrix can get messy so it can be clearer to write  and then find expressions for a, b, c, d on separate lines

• STEP 4: The conclusion step
  • State the result is true
  • Explain in words why the result is true
  • It must include:
    • If true for n = k then it is true for n = k + 1
    • Since true for n = 1 the statement is true for all n ∈ ℤ, n ≥ 1  by mathematical induction
  • The sentence will be the same for each proof just change the base case from n = 1 if necessary