Lesson Interior angles of a polygon and Arithmetic progression
Algebra
->
Sequences-and-series
-> Lesson Interior angles of a polygon and Arithmetic progression
Log On
Algebra: Sequences of numbers, series and how to sum them
Section
Solvers
Solvers
Lessons
Lessons
Answers archive
Answers
Source code of 'Interior angles of a polygon and Arithmetic progression'
This Lesson (Interior angles of a polygon and Arithmetic progression)
was created by by
ikleyn(52781)
:
View Source
,
Show
About ikleyn
:
<H2>Interior angles of a polygon and Arithmetic progression</H2> <H3>Problem 1</H3>The interior angles of a convex nonagon have degree measures that are integers in arithmetic sequence. What is the smallest angle possible in this nonagon? <B>Solution</B> <pre> From one side, the sum of interior angles of any 9-gon is (9-2)*180 = 7*180 degrees. From the other side, the sum of the first n terms of any arithmetic progression is {{{S[n]}}} = {{{((a[1]+a[n])/2)*n}}}. In our case, {{{((a[1]+a[9])/2)*9}}} = 7*180, so {{{(a[1]+a[9])/2}}} = 140 degrees. For arithmetic progression, the average of any two terms, equally remoted from the central term, is the same. In particular, the central term {{{a[5]}}} is 140 degrees. The terms of the AP are {{{a[4]}}} = 140 - d, {{{a[6]}}} = 140 + d, {{{a[3]}}} = 140 - 2d, {{{a[7]}}} = 140 + 2d, {{{a[2]}}} = 140 - 3d, {{{a[8]}}} = 140 + 3d, {{{a[1]}}} = 140 - 4d, {{{a[9]}}} = 140 + 4d. To make {{{a[1]}}} as small as possible, we should take the common difference as large as possible. We have two constraints: {{{a[1]}}} must be positive, {{{a[1]}}} > 0, (1) and {{{a[9]}}} must be less than 180 degrees; so {{{a[9]}}} must be 176 degrees. (2) Of these two constraints, the constraint (2) is more cumbersome, and it gives d = 36/4 = 9 degrees. Then both constraints (1) and (2) are satisfied. Thus the minimum angle is {{{a[1]}}} = 140 - 4*9 = 140 - 36 = 104 degrees. <U>ANSWER</U> </pre> My other lessons on arithmetic progressions in this site are - <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Arithmetic-progressions.lesson>Arithmetic progressions</A> - <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/The-proofs-of-the-formulas-for-arithmetic-progressions.lesson>The proofs of the formulas for arithmetic progressions</A> - <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Problems-on-arithmetic-progressions.lesson>Problems on arithmetic progressions</A> - <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Word-problems-on-arithmetic-progressions.lesson>Word problems on arithmetic progressions</A> - <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Chocolate-bars-and-arithmetic-progressions.lesson>Chocolate bars and arithmetic progressions</A> - <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Free-fall-and-arithmetic-progressions.lesson>Free fall and arithmetic progressions</A> - <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Uniformly-accelerated-motions-and-arithmetic-progressions.lesson>Uniformly accelerated motions and arithmetic progressions</A> - <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Increments-of-a-quadratic-function-form-an-arithmetic-progression.lesson>Increments of a quadratic function form an arithmetic progression</A> - <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/One-characteristic-property-of-arithmetic-progressions.lesson>One characteristic property of arithmetic progressions</A> - <A HREF=https://www.algebra.com/algebra/homework/Sequences-and-series/Solved-problems-on-arithmetic-progressions.lesson>Solved problems on arithmetic progressions</A> - <A HREF=https://www.algebra.com/algebra/homework/Sequences-and-series/Calculating-partial-sums-of-arithmetic-progressions.lesson>Calculating partial sums of arithmetic progressions</A> - <A HREF=https://www.algebra.com/algebra/homework/Sequences-and-series/Finding-number-of-terms-of-an-arithmeti--progression.lesson>Finding number of terms of an arithmetic progression</A> - <A HREF=https://www.algebra.com/algebra/homework/Sequences-and-series/Inserting-arithmetic-means-between-given-numbers.lesson>Inserting arithmetic means between given numbers</A> - <A HREF=https://www.algebra.com/algebra/homework/Sequences-and-series/Advanced-problems-on-arithmetic-progressions.lesson>Advanced problems on arithmetic progressions</A> - <A HREF=https://www.algebra.com/algebra/homework/Sequences-and-series/Marh-Olimpiad-level-problem-on-arithmetic-progression.lesson>Math Olympiad level problem on arithmetic progression</A> - <A HREF=https://www.algebra.com/algebra/homework/Sequences-and-series/Problems-on-arithmetic-progressions-solved-MENTALLY.lesson>Problems on arithmetic progressions solved MENTALLY</A> - <A HREF=https://www.algebra.com/algebra/homework/Sequences-and-series/Entertainment-problems-on-arithmetic-progressions.lesson>Entertainment problems on arithmetic progressions</A> - <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Mathematical-induction-and-arithmetic-progressions.lesson>Mathematical induction and arithmetic progressions</A> - <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> OVERVIEW of my lessons on arithmetic progressions with short annotations is in the lesson <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Review-of-lessons-on-arithmetic-progressions.lesson>OVERVIEW of lessons on arithmetic progressions</A>. Use this file/link <A HREF=https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-II - YOUR ONLINE TEXTBOOK</A> to navigate over all topics and lessons of the online textbook ALGEBRA-II.
