Lesson Mathematical induction and geometric progressions

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Mathematical induction and geometric progressions


The formulas for  n-th  term of a geometric progression and for sum of the first  n  terms of a geometric progression were just proved in the lesson
The proofs of the formulas for geometric progressions  under the current topic in this site.

You will learn from  this  lesson how to prove these formulas using the method of  Mathematical Induction.

The method of  Mathematical Induction  was explained in the lessons  Mathematical induction and arithmetic progressions  under the current topic in this site.
Let me remind you this method.

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|    Let  S(n)  be a mathematical statement which relates to any natural number of the infinite sequence  n = 1, 2, 3, . . .
|    Two steps should be done to prove this statement by the method of mathematical induction:
|        1) We have to prove that the statement  S(1)  is valid.
|        2) We have to prove next implication.
|              If the statement  S(k)  is true, then the statement  S(k+1)  is true,  where  k  is any positive integer number.
|    If these two steps are done, then the statement  S(n)  is proved for all positive integer numbers  n.
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Problem 1

Let  a%5B1%5D, a%5B1%5D, a%5B2%5D,  . . . ,  a%5Bn%5D, . . . ,  be a geometric progression with the first term  a%5B1%5D  and the ratio  q.
Prove the formula for n-th term of a geometric progression

a%5Bn%5D = a%5B1%5D%2Aq%5E%28n-1%29                                                                                                  (1)

by the method of Mathematical Induction.

Solution
First, let us check the formula  (1)  for  n=1.  Both the left and the right side of this formula is equal to the value of  a%5B1%5D  at  n=1.  So, the base of the Mathematical Induction method is true.

Now, let us prove the implication step of the Mathematical Induction.  In other words, let us suppose that the formula  (1)  is valid for some positive integer  k:

a%5Bk%5D = a%5B1%5D%2Aq%5E%28k-1%29.                                                                                                 (2)

According to the definition of the geometric progression  (see the lesson  Geometric progressions  in this site)

a%5Bk%2Bl%5D = a%5Bk%5D%2Aq.                                                                                                     (3)

From the formulas (2) and (3) you have

a%5Bk%2Bl%5D = a%5B1%5D%2Aq%5Ek.                                                                                                   (4)

The formula  (4)  is exactly the formula  (2)  applied to the next integer  k+1  instead of  k.  So, we proved that if the formula  (2)  is true for the positive integer  k,
then it is true for the next positive integer  k+1. The step of induction is completed.

According to the  Principle of Mathematical Induction  the formula  (1)  is proved for all positive integer  n.


Problem 2

Let  a%5B1%5D, a%5B1%5D, a%5B2%5D,  . . . ,  a%5Bn%5D, . . . ,  be a geometric progression with the first term  a%5B1%5D  and the ratio  q.
Prove the formula for sum of the first n terms of a geometric progression

a%5B1%5D + a%5B1%5Dq + a%5B1%5Dq%5E2 + . . . + a%5B1%5Dq%5E%28n-1%29 = a%5B1%5D%2A%28q%5En-1%29%2F%28q-1%29,                                             (5)

by the method of Mathematical Induction.

Solution
First, let us check the formula  (5)  for  n=1.  Both the left and the right side of this formula is equal to the value of  a%5B1%5D  at  n=1.  So, the base of the Mathematical Induction method is true.

Now, let us prove the implication step of the Mathematical Induction.  In other words, let us suppose that the formula  (1)  is valid for some positive integer  k:

a%5B1%5D + a%5B1%5Dq + a%5B1%5Dq%5E2 + . . . + a%5B1%5Dq%5E%28k-1%29 = a%5B1%5D%2A%28q%5Ek-1%29%2F%28q-1%29,                                             (6)

Consider the sum for the next integer  k+1 

a%5B1%5D + a%5B1%5Dq + a%5B1%5Dq%5E2 + . . . + a%5B1%5Dq%5E%28k-1%29 + a%5B1%5Dq%5Ek

Let us group the first  k  terms by placing them into the square brackets.  Our sum is equal to

[a%5B1%5D + a%5B1%5Dq + a%5B1%5Dq%5E2 + . . . + a%5B1%5Dq%5E%28k-1%29] + a%5B1%5Dq%5Ek.                                                  (7)

Now you can replace the sum of the first  k  additives by  a%5B1%5D%2A%28q%5Ek-1%29%2F%28q-1%29  based on our assumption  (6).  Hence, the sum  (7)  is equal to
[a%5B1%5D + a%5B1%5Dq + a%5B1%5Dq%5E2 + . . . + a%5B1%5Dq%5E%28k-1%29] + a%5B1%5Dq%5Ek =  a%5B1%5D%2A%28q%5Ek-1%29%2F%28q-1%29  +  a%5B1%5Dq%5Ek.              (8)

The right side of  (8)  is equal to

a%5B1%5D%2A%28q%5Ek-1%29%2F%28q-1%29  +  a%5B1%5Dq%5Ek = %28a%5B1%5Dq%5Ek+-+a%5B1%5D+%2B+a%5B1%5Dq%5E%28k%2B1%29+-+a%5B1%5Dq%5Ek%29%2F%28q-1%29 = %28a%5B1%5Dq%5E%28k%2B1%29+-+a%5B1%5D%29%2F%28q-1%29 = a%5B1%5D%2A%28q%5E%28k%2B1%29+-+1%29%2F%28q-1%29.              (9)

From the formulas  (8)  and  (9)  you have

a%5B1%5D + a%5B1%5Dq + a%5B1%5Dq%5E2 + . . . + a%5B1%5Dq%5E%28k-1%29 + a%5B1%5Dq%5Ek = a%5B1%5D%2A%28q%5E%28k%2B1%29+-+1%29%2F%28q-1%29.                          (10)

The formula  (10)  is exactly the formula  (6)  applied to the next integer  k+1  instead of  k.  So, we proved that if the formula  (6)  is true for the positive integer  k,
then it is true for the next positive integer  k+1. The step of induction is completed.

According to the  Principle of Mathematical Induction  the formula  (5)  is proved for all positive integer  n.


My lessons on geometric progressions in this site are
    - Geometric progressions
    - The proofs of the formulas for geometric progressions
    - Problems on geometric progressions
    - Solved problems on geometric progressions
    - Word problems on geometric progressions
    - One characteristic property of geometric progressions
    - Ordinary Annuity saving plans and geometric progressions
    - Annuity Due saving plans and geometric progressions
    - Solved problems on Ordinary Annuity saving plans
    - Miscellaneous problems on Annuity saving plans
    - Fresh, sweet and crispy problem on arithmetic and geometric progressions
    - Mathematical induction and geometric progressions                                                                   (this lesson)
    - Mathematical induction for sequences other than arithmetic or geometric
    - OVERVIEW of lessons on geometric progressions.

My lessons on the Method of Mathematical induction in this site are
    - Mathematical induction and arithmetic progressions
    - Mathematical induction and geometric progressions                                                                   (this lesson)
    - Mathematical induction for sequences other than arithmetic or geometric
    - Proving inequalities by the method of Mathematical Induction
    - OVERVIEW of lessons on the Method of Mathematical induction

Use this file/link  ALGEBRA-II - YOUR ONLINE TEXTBOOK  to navigate over all topics and lessons of the online textbook  ALGEBRA-II.

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