Lesson Mathematical induction and geometric progressions

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<H2>Mathematical induction and geometric progressions</H2>

The formulas for &nbsp;n-th &nbsp;term of a geometric progression and for sum of the first &nbsp;n&nbsp; terms of a geometric progression were just proved in the lesson 
<A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/The-proofs-of-the-formulas-for-geometric-progressions.lesson>The proofs of the formulas for geometric progressions</A>&nbsp; under the current topic in this site.

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

The method of &nbsp;Mathematical Induction&nbsp; was explained in the lessons &nbsp;<A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Mathematical-induction-and-arithmetic-progressions.lesson>Mathematical induction and arithmetic progressions</A>&nbsp; under the current topic in this site. 
Let me remind you this method.

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|&nbsp;&nbsp;&nbsp;&nbsp;Let &nbsp;S(n)&nbsp; be a mathematical statement which relates to any natural number of the infinite sequence &nbsp;n = 1, 2, 3, . . . 
|&nbsp;&nbsp;&nbsp;&nbsp;Two steps should be done to prove this statement by the method of mathematical induction:
|&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;1) We have to prove that the statement &nbsp;S(1)&nbsp; is valid.
|&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;2) We have to prove next implication. 
|&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;If the statement &nbsp;S(k)&nbsp; is true, then the statement &nbsp;S(k+1)&nbsp; is true, &nbsp;where &nbsp;k&nbsp; is any positive integer number.
|&nbsp;&nbsp;&nbsp;&nbsp;If these two steps are done, then the statement &nbsp;S(n)&nbsp; is proved for all positive integer numbers &nbsp;n.
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</BLOCKQUOTE><H3>Problem 1</H3>Let &nbsp;{{{a[1]}}}, {{{a[1]}}}, {{{a[2]}}}, &nbsp;. . . , &nbsp;{{{a[n]}}}, . . . , &nbsp;be a geometric progression with the first term &nbsp;{{{a[1]}}}&nbsp; and the ratio &nbsp;{{{q}}}. 
Prove the formula for n-th term of a geometric progression

{{{a[n]}}} = {{{a[1]*q^(n-1)}}} &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(1)

by the method of Mathematical Induction.

Solution

First, let us check the formula &nbsp;(1)&nbsp; for &nbsp;n=1. &nbsp;Both the left and the right side of this formula is equal to the value of &nbsp;{{{a[1]}}}&nbsp; at &nbsp;n=1.&nbsp; So, the base of the Mathematical Induction method is true.

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

{{{a[k]}}} = {{{a[1]*q^(k-1)}}}. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(2)

According to the definition of the geometric progression &nbsp;(see the lesson &nbsp;<A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Geometric-progressions.lesson>Geometric progressions</A>&nbsp; in this site)

{{{a[k+l]}}} = {{{a[k]*q}}}. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(3)

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

{{{a[k+l]}}} = {{{a[1]*q^k}}}. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(4)

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

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

<H3>Problem 2</H3>Let &nbsp;{{{a[1]}}}, {{{a[1]}}}, {{{a[2]}}}, &nbsp;. . . , &nbsp;{{{a[n]}}}, . . . , &nbsp;be a geometric progression with the first term &nbsp;{{{a[1]}}}&nbsp; and the ratio &nbsp;{{{q}}}. 
Prove the formula for sum of the first n terms of a geometric progression

{{{a[1]}}} + {{{a[1]q}}} + {{{a[1]q^2}}} + . . . + {{{a[1]q^(n-1)}}} = {{{a[1]*(q^n-1)/(q-1)}}}, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(5)

by the method of Mathematical Induction.

Solution

First, let us check the formula &nbsp;(5)&nbsp; for &nbsp;n=1. &nbsp;Both the left and the right side of this formula is equal to the value of &nbsp;{{{a[1]}}}&nbsp; at &nbsp;n=1.&nbsp; So, the base of the Mathematical Induction method is true.

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

{{{a[1]}}} + {{{a[1]q}}} + {{{a[1]q^2}}} + . . . + {{{a[1]q^(k-1)}}} = {{{a[1]*(q^k-1)/(q-1)}}}, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(6)

Consider the sum for the next integer &nbsp;k+1&nbsp;

{{{a[1]}}} + {{{a[1]q}}} + {{{a[1]q^2}}} + . . . + {{{a[1]q^(k-1)}}} + {{{a[1]q^k}}}

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

[{{{a[1]}}} + {{{a[1]q}}} + {{{a[1]q^2}}} + . . . + {{{a[1]q^(k-1)}}}] + {{{a[1]q^k}}}.&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(7)

Now you can replace the sum of the first &nbsp;k&nbsp additives by &nbsp;{{{a[1]*(q^k-1)/(q-1)}}}&nbsp; based on our assumption &nbsp;(6). &nbsp;Hence, the sum &nbsp;(7)&nbsp; is equal to

[{{{a[1]}}} + {{{a[1]q}}} + {{{a[1]q^2}}} + . . . + {{{a[1]q^(k-1)}}}] + {{{a[1]q^k}}} = &nbsp;{{{a[1]*(q^k-1)/(q-1)}}}&nbsp; + &nbsp;{{{a[1]q^k}}}. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(8)

The right side of &nbsp;(8)&nbsp; is equal to

{{{a[1]*(q^k-1)/(q-1)}}}&nbsp; + &nbsp;{{{a[1]q^k}}} = {{{(a[1]q^k - a[1] + a[1]q^(k+1) - a[1]q^k)/(q-1)}}} = {{{(a[1]q^(k+1) - a[1])/(q-1)}}} = {{{a[1]*(q^(k+1) - 1)/(q-1)}}}.&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(9)

From the formulas &nbsp;(8)&nbsp; and &nbsp;(9)&nbsp; you have

{{{a[1]}}} + {{{a[1]q}}} + {{{a[1]q^2}}} + . . . + {{{a[1]q^(k-1)}}} + {{{a[1]q^k}}} = {{{a[1]*(q^(k+1) - 1)/(q-1)}}}.&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(10)

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

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

My lessons on geometric progressions in this site are

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Geometric-progressions.lesson>Geometric progressions</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/The-proofs-of-the-formulas-for-geometric-progressions.lesson>The proofs of the formulas for geometric progressions</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Problems-on-geometric-progressions.lesson>Problems on geometric progressions</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Sequences-and-series/Solved-problems-on-geometric-progressions.lesson>Solved problems on geometric progressions</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Word-problems-on-geometric-progressions.lesson>Word problems on geometric progressions</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/One-characteristic-property-of-geometric-progressions.lesson>One characteristic property of geometric progressions</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Ordinary-Annuity-saving-plans-and-geometric-progressions.lesson>Ordinary Annuity saving plans and geometric progressions</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Annuity-due-saving-plans-and-geometric-progressions.lesson>Annuity Due saving plans and geometric progressions</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Sequences-and-series/Solved-problem-on-Ordinary-Annuity-saving-plans.lesson>Solved problems on Ordinary Annuity saving plans</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Sequences-and-series/Solved-problems-on-Annuity-saving-plans.lesson>Miscellaneous problems on Annuity saving plans</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Sequences-and-series/Fresh-sweet-and-crispy-problem-on-arithmetic-and-geometric-progressions.lesson>Fresh, sweet and crispy problem on arithmetic and geometric progressions</A>

&nbsp;&nbsp;&nbsp;&nbsp;- Mathematical induction and geometric progressions&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(this lesson)

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Mathematical-induction-for-sequences-other-than-arithmetic-or-geometric.lesson>Mathematical induction for sequences other than arithmetic or geometric</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Review-of-lessons-on-geometric-progressions.lesson>OVERVIEW of lessons on geometric progressions</A>.

My lessons on the Method of Mathematical induction in this site are

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Mathematical-induction-and-arithmetic-progressions.lesson>Mathematical induction and arithmetic progressions</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Proving-inequalities-by-the-method-of-Mathematical-Induction.lesson>Proving inequalities by the method of Mathematical Induction</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Sequences-and-series/OVERVIEW-of-lessons-on-the-Method-of-Mathematical-induction.lesson>OVERVIEW of lessons on the Method of Mathematical induction</A>

Use this file/link &nbsp;<A HREF=https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-II - YOUR ONLINE TEXTBOOK</A>&nbsp; to navigate over all topics and lessons of the online textbook &nbsp;ALGEBRA-II.