Lesson OVERVIEW of lessons on arithmetic progressions

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OVERVIEW of lessons on arithmetic progressions


An arithmetic progression is the sequence of numbers with a constant difference between the current term and the preceding term for any two consecutive terms.

The n-th term of an arithmetic progression  a%5B1%5D, a%5B2%5D, a%5B3%5D, ..., a%5Bn%5D  with the first term  a%5B1%5D  and the common difference  d  is
    a%5Bn%5D = a%5B1%5D+%2B+%28n-1%29%2Ad.

The sum of n terms of an arithmetic progression  a%5B1%5D, a%5B2%5D, a%5B3%5D, ..., a%5Bn%5D  with the first term  a%5B1%5D  and the common difference  d  is equal to
    S%5Bn%5D = a%5B1%5D+%2B+a%5B2%5D+%2B+a%5B3%5D+%2B+ellipsis+%2B+a%5Bn%5D = %28a%5B1%5D+%2B+%28n-1%29%2Ad%2F2%29%2An.

Another formula for the sum of the first  n  terms of an arithmetic progression  a%5B1%5D, a%5B2%5D, a%5B3%5D, ..., a%5Bn%5D  is
    S%5Bn%5D = %28a%5B1%5D+%2B+a%5Bn%5D%29%2F2%2An

The formula for the sum of the first  n  natural numbers is

1 + 2 + 3 + ... + n = n%2A%28n%2B1%29%2F2.

The formula for the sum of the first  n   odd   positive integer numbers is

1 + 3 + 5 + ... + (2n-1) = n%5E2.

The formula for the sum of the first  n   even   positive integer numbers is

2 + 4 + 6 + ... + 2n = n%2A%28n%2B1%29.


For the proofs of these facts see the lessons
    Arithmetic progressions and
    The proofs of the formulas for arithmetic progressions
under the current topic in this site.


For problems on arithmetic progressions see the lessons
    Problems on arithmetic progressions,
    Word problems on arithmetic progressions
    Chocolate bars and arithmetic progressions
    One characteristic property of arithmetic progressions
    Solved problems on arithmetic progressions
    Calculating partial sums of arithmetic progressions
    Finding number of terms of an arithmetic progression
    Inserting arithmetic means between given numbers
    Advanced problems on arithmetic progressions
    Interior angles of a polygon and Arithmetic progression
    Problems on arithmetic progressions solved MENTALLY
    Entertainment problems on arithmetic progressions
under the current topic in this site.

You will find problems  on pipes stacked in a pile;  on four years old boy playing with Lego bricks;  on chocolate bars stacked on a shelf in a grocery store;  on rows
of seats in a concert hall,  and more.


There are interesting, not widely known and unexpecting facts related to arithmetic progressions:
The distances that the free falling body falls during the first second, the next one, the third second and so on, form the arithmetic progression.

The distances that an uniformly accelerated body covers during the first second, the next one, the third second and so on, form
an arithmetic progression.

The increments of a quadratic polynomial function over the sequence of uniformly distributed points form an arithmetic progression.

For the proofs of these facts see the lessons
    Free fall and arithmetic progressions,
    Uniformly accelerated motions and arithmetic progressions and
    Increments of a quadratic function form an arithmetic progression,
under the current topic in this site.

There is a special method for proving facts on sequences, the  Mathematical Induction method.  You will find a description of this method and examples that show
how it works in the lesson  Mathematical induction and arithmetic progressions  under the current topic in this site.


Arithmetic progressions have the characteristic property that differentiates them from other sequences.

        A sequence  a%5B1%5D, a%5B2%5D, a%5B3%5D, . . . , a%5Bn%5D  is an arithmetic progression if and only if for any three consecutive terms  a%5Bk-l%5D,  a%5Bk%5D,  a%5Bk%2Bl%5D
        the middle term is equal to the half-sum of its neighbors: a%5Bk%5D = %28a%5Bk-l%5D%2Ba%5Bk%2Bl%5D%29%2F2.
For the proof of this property see the lesson  One characteristic property of arithmetic progressions  under the current topic in this site.


The lesson  Mathematical induction for sequences other than arithmetic or geometric  teach you on proving formulas by the method of Mathematical Induction
for the sequences other than arithmetic or geometric progressions.

You will find the proofs of the following formulas there:

1%5E2 + 2%5E2 + 3%5E2 + . . . + n%5E2 = %28n%2A%28n%2B1%29%2A%282n%2B1%29%29%2F6,

1%5E3 + 2%5E3 + 3%5E3 + . . . + n%5E3 = %28n%5E2%2A%28n%2B1%29%5E2%29%2F4,

%281-1%2F4%29 * %281-1%2F9%29 * %281-1%2F16%29 * . . . * %281-1%2F%28n%5E2%29%29 = %28n%2B1%29%2F%282n%29,

and

1%2F%281%2A2%29 + 1%2F%282%2A3%29 + 1%2F%283%2A4%29 + . . . + 1%2F%28n%2A%28n%2B1%29%29 = n%2F%28n%2B1%29.



The lesson  Problems on arithmetic progressions  contains the solutions to these problems:

    Problem 1.  Derive the formula for the sum of the first  n  natural numbers.

    Problem 2.  Derive the formula for the sum of the first  n   odd   positive integer numbers.

    Problem 3.  Derive the formula for the sum of the first  n   even   positive integer numbers.

    Problem 4.  Find the  21-th  term of the arithmetic progression if its  11-th  term is  53  and the  17-th  term is  71.

    Problem 5.  Find the sum of the first  31  terms of the arithmetic progression if its  7-th  term is equal to  21  and the  11-th  term is equal to  29.


The lesson  Word problems on arithmetic progressions   contains the solutions to these problems:

    Problem 1.  Pipes are stacked in a pile as shown in the figure.  The bottom row has  11  pipes.
                       There are totally  6  rows of pipes in the pile.  How many pipes are stacked in the pile?

    Problem 2.  Four years old  Michael plays with Lego bricks.  He wants  to built the construction shown in the figure,
                       with  4  bricks at the bottom.  How many  Lego bricks does he need?

    Problem 3.  This problem is very similar to Problem 2.

    Problem 4.  There are  20  rows of seats in a concert hall with  20  seats in the first row,
                       21  seats in the second row,  22  seats in the third row,  and so on.
                       In total,  how many seats are there in the concert hall?

    Problem 5.  An oil company bores a hole  80 m deep.  Estimate the cost of boring if the cost is  $30  for drilling the first meter
                       with an increase in cost of  $2  per meter for each succeeding meter.


The lesson  One characteristic property of arithmetic progressions  contains the solutions to these problems:

    Problem 1.  Let  a%5B1%5D, a%5B2%5D, a%5B3%5D, . . . , a%5Bn%5D  be an arithmetic progression.
                       Prove that for any three consecutive terms  a%5Bk-l%5D,  a%5Bk%5D and  a%5Bk%2Bl%5D  the middle term is equal to the half-sum of its neighbors
                               a%5Bk%5D = %28a%5Bk-l%5D%2Ba%5Bk%2Bl%5D%29%2F2.

    Problem 2.  Let  a%5B1%5D, a%5B2%5D, a%5B3%5D, . . . , a%5Bn%5D  be a sequence such that for any three consecutive terms  a%5Bk-l%5D,  a%5Bk%5D and  a%5Bk%2Bl%5D  the middle term is equal to
                       the half-sum of its neighbors:     a%5Bk%5D = %28a%5Bk-l%5D+%2B+a%5Bk%2Bl%5D%29%2F2.     Prove that this sequence is an arithmetic progression.

    Problem 3.  The first three terms of an arithmetic progression are  2sin(x),  3cos(x)  and  (sin(x) + 2cos(x)),  respectively,  where  x  is an acute angle.
                       Show that  tan(x) =  4/3.  Hence,  find the sum of the first twenty terms of the progression.


The lesson  Solved problems on arithmetic progressions  contains the solutions to nine problems:

    Problem 1.  An arithmetic progression consists of three terms whose sum is  48  and the sum of their squares is  800.
                       Find the progression.

    Problem 2.  If the  6-th term of an arithmetic progression is  121,  find the sum of the first  11  terms.

    Problem 3.  The sum of the first  5  terms of an AP is  30  and the fourth term is  44.  Find the common difference and the sum of the first 10 terms.

    Problem 4.  If the fifth term of arithmetic sequence is  23  and the sum of the first ten terms of the sequence is  240,
                       then find the sum of the first sixty terms of this sequence.

    Problem 5.  The sum of the fifth and seventh terms of an arithmetic series is  38,  while the sum of the first fifteen terms is  375.
                       Determine the first term and the common difference.

    Problem 6.  If  7  times the  7-th term of an AP is equal to  11  times its  11-th term,  show that its  18-th term is  0.

    Problem 7.  The sum of the first five terms of an arithmetic progression is  10,  the sum of their squares is  380.
                       Find the first term and common difference.

    Problem 8.  The coefficients  a,  b,  c of the quadratic equation  ax%5E2%2Bbx%2Bc = 0  form an arithmetic progression.  One root of this equation is  2.
                       Find the other root.

    Problem 9.  The first three terms of an arithmetic progression are  x,  2x+1  and  5x+1.  Find  x  and the sum of the first  10  terms.


The lesson  Calculating partial sums of arithmetic progressions  contains the solutions to these problems:

    Problem 1.  In the sequence of  99  first natural numbers from  1  to  99  inclusive,
                       each third term is excluded.  Find the sum of remaining numbers.

    Problem 2.  Find the sum of all even numbers from  4  to  100  inclusive,  excluding those which are multiples of  3.


The lesson     Finding number of terms of an arithmetic progression  contains the solutions to these problems:

    Problem 1.  Find the number of terms in sequence  12,  7,  2, . . . ,  -203.

    Problem 2.  The partial sum in the arithmetic series with first term  17  and a common difference of  3  is  30705.
                       How many terms are there in the series?

    Problem 3.  An arithmetic sequence  a%5B1%5D, a%5B2%5D, . . . , a%5B100%5D  has a sum of  15000.  Find the first term
                       and the common difference,  if the sum of the terms in the sequence  a%5B3%5D, a%5B6%5D, a%5B9%5D, . . . , a%5B99%5D  is  5016.

    Problem 4.  How many terms has an arithmetic progression where sum is  1340  with first term  3  and  n-th  term  131 ?


The lesson  Inserting arithmetic means between given numbers  contains the solution to these problems:

    Problem 1.  Insert seven arithmetic means between  2  and  26.

    Problem 2.  Insert eight arithmetic mean between  26  and  -10.


The lesson  Advanced problems on arithmetic progressions  contains the solution to these problems:

    Problem 1.  Find six numbers in  AP,  such that the sum of the two extremes be  16  and the product of the two middle terms be  63.

    Problem 2.  In the following two AP's how many terms are identical?  2, 5, 8, 11 . . .  to  60 terms,  and  3, 5, 7 . . .  50 terms.

    Problem 3.  Two arithmetic progressions have the same first and last terms.
                       The first  AP  has  21  terms with a common difference of  9.  How many terms has the other  AP  if its common difference is  4?

    Problem 4.  In an  AP  consisting of  15 terms,  the sum of the last five terms is  305.
                       If the sixth term is  26,  find the sum of this  AP.

    Problem 5.  If  p  and  q  are the fourth and the seventh terms respectively of an arithmetic series,
                           1)   determine the common difference,  and
                           2)   find the sum of the ten terms of the series.

    Problem 6.  There are four successive terms in an  Arithmetic progression.
                       The sum of the two extremes is  8,  and the product of the middle two numbers is  15.  What are the numbers?

    Problem 7.  Find four numbers in  AP  whose sum is  20  and the sum of squares is  120.

    Problem 8.  The sum of  3  terms of an arithmetic progression is  42  and the product of the first and third terms is  52.
                       Find the  3  terms.

    Problem 9.  Find three consecutive positive odd integers such that the sum of their squares is  371.

    Problem 10.  The sum of  40  terms of a certain arithmetic sequence is  430,  while the sum of  60  terms is  945.
                         Determine the first term and the common difference of the arithmetic sequence.

    Problem 11.  What is the first of  100  consecutive odd integers whose sum is   100%5E100 ?

    Problem 12.  The pattern forming the irrational number   0.12340432100123400043210000...   continues indefinitely.
                       What is the  1945-th digit after the decimal dot in this pattern?


The lesson  Interior angles of a polygon and Arithmetic progression  contains the solution to this problem

    Problem 1.  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?


The lesson  Problems on arithmetic progressions solved MENTALLY  contains the solutions to these problems:

    Problem 1.  In arithmetic progression,  the first term is  104;  the n-th term is  680  and the common difference is  16.  Find  n.

    Problem 2.  Find the sum   75 + 72 + 69 + 66 + . . . + 21 + 18 + 15.

    Problem 3.  Find the sum of the first  15  terms of an arithmetic sequence with the first term  -4  and the last term  66.

    Problem 4.  Find the sum of  15  terms of an arithmetic series if the middle term is  12.

    Problem 5.  Find the first term of the  AP,  if the  2p-th term is  50 more than the  p-th term,  and the  (p+1)-th term is  56.

    Problem 6.  If The sum of the  17  first terms of an arithmetic progression is  629  and its first term is  -3,  find the common
                       difference of the  AP  and its  29-th term.

    Problem 7.  James decided to start exercising.  The first day he jogged for  10  minutes.  The next day he wanted to jog for  4  more minutes
                       than the previous day.  He wanted to continue increasing by the same amount like this for every one of the days in a week.
                       If he succeeded in doing this,  how many total minutes did he run for the week?

    Problem 8.  If  f(1) = 9  and  f(n) = f(n-1)+2,  then find the value of  f(6).

    Problem 9.  David jogged for  5  days in a row.  Each day he jogged  1.4 km more than the day before.
                       At the end of  5  days he jogged a total of  20.6 km.  How far did he jogged on the  5th day.


The lesson  - Entertainment problems on arithmetic progressions  contains the solution to these problems:

    Problem 1.  Express as a fraction in simplest form   %288+%2B+16+%2B+24+%2B+ellipsis+%2B+784%29%2F%286+%2B+12+%2B+18+%2B+ellipsis+%2B+588%29.

    Problem 2.  Without computing each sum, find which is greater,  "O"  or  "E" and by how much
                           O = 5 + 7 + 9 + 11 + . . + 105,   E = 4 + 6 + 8 + 10 + . . . + 104.

    Problem 3.  A besieged fortress is held by  5700  men who have provisions for  66  days.
                       Of the garrison loses  20  men each day,  for how many days will be provisions last?

    Problem 4.  If  400  more than the sum of  p  consecutive integers is equal
                       to the sum of the next  p  consecutive integers,  what is the value of  p?

    Problem 5.  Consider an arithmetic sequence  7,  5,  3, . . .
                       For which integer value of  n  is the  n-th term equal to the sum of the terms to that point
                       in this sequence ?

    Problem 6.  A tight roll of paper as delivered to the printer is  60 cm in diameter and the paper
                       is wound onto a wooden cylinder  8 cm in diameter.  The paper is  0.005 cm thick.
                       What length of paper is there in the roll in kilometers?

    Problem 7.  A roll of tape is on a spool with diameter  3.6 cm.  The diameter of the spool plus the diameter
                       of the tape is  5.4 cm.  The tape is  25.4 m long.  Find the number of layers on the tape.


The lesson  Math Olympiad level problems on arithmetic progression  contains the solution to these problems:

    Problem 1.  A series of  384  consecutive odd integers has a sum that is a perfect fourth power of a positive integer.
                       Find the smallest possible sum for this series.

    Problem 2.  A series of  567  consecutive positive integers has a sum that is a perfect cube.
                       Find the smallest possible positive sum for this series.


The full list of my lessons on arithmetic progressions in this site is
    - Arithmetic progressions
    - The proofs of the formulas for arithmetic progressions
    - Problems on arithmetic progressions
    - Word problems on arithmetic progressions
    - Chocolate bars and arithmetic progressions
    - Free fall and arithmetic progressions
    - Uniformly accelerated motions and arithmetic progressions
    - Increments of a quadratic function form an arithmetic progression
    - One characteristic property of arithmetic progressions
    - Solved problems on arithmetic progressions
    - Calculating partial sums of arithmetic progressions
    - Finding number of terms of an arithmetic progression
    - Inserting arithmetic means between given numbers
    - Advanced problems on arithmetic progressions
    - Interior angles of a polygon and Arithmetic progression
    - Math Olympiad level problems on arithmetic progression
    - Problems on arithmetic progressions solved MENTALLY

    - Entertainment problems on arithmetic progressions

    - Mathematical induction and arithmetic progressions
    - Mathematical induction for sequences other than arithmetic or geometric

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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