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Edexcel A Level Further Maths: Core Pure:復(fù)習(xí)筆記9.1.2 Common Cases of Proof by Induction

Category: A-level課程, 國(guó)際課程, 教材筆記, 福利干貨 Date: 2022年10月21日 下午3:37

Proof by Induction - Sequences

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?∈ ?,?n?≥ 1 ?by mathematical induction
    • The sentence will be the same for each proof just change the base case from?n?= 1 if necessary

Worked Example

A sequence is defined by .

Prove by mathematical induction that .

9-1-2-edexcel-al-fm-cp-proof-by-induction-sequences-we

 

Proof by Induction - Series

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?∈ ?,?n?≥ 1 ?by mathematical induction
    • The sentence will be the same for each proof just change the base case from?n?= 1 if necessary

Worked Example

Prove by induction that .

1-5-1-ib-aa-hl-proof-by-induction-we-solution

 

Proof by Induction - Divisibility

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?∈ ?,?n?≥ 1 ?by mathematical induction
    • The sentence will be the same for each proof just change the base case from?n?= 1 if necessary

Worked Example

Prove by induction that is divisible by 7 for all integers .

9-1-2-edexcel-al-fm-cp-proof-by-induction-divisibility-we

 

Proof by Induction - Matrices

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?∈ ?,?n?≥ 1 ?by mathematical induction
    • The sentence will be the same for each proof just change the base case from?n?= 1 if necessary

Worked Example

The matrix ?is given by .

Prove, using mathematical induction, that .

9-1-2-edexcel-al-fm-cp-proof-by-induction-matrices-we

 

 

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