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This lesson on arithmetic sequences and series (1 + 2 + 3 + … + 98 + 99 + 100 = ?) might improve SAT scores

Subject:

Math  

Grade:

10  

Title – Arithmetic Series
By – Romeo Poralan
Primary Subject – Math
Grade Level – 10
Time Frame – 60 minutes

Reference:

      “Exploring Mathematics II (Intermediate Algebra)”

    By Orlando A. Oronce and Marilyn O. Mendoza

Objectives:

    At the end of the learning period, at least 75% of the students with at least 75% proficiency will be able to:

    • differentiate between an arithmetic sequence and an arithmetic series
    • solve for the sum of the arithmetic sequence without writing all the terms or using the longhand method.

Procedure:

  • Motivation:
    1. Play a game — Mathematical treasure-hunt
      • Each student is given a sheet of paper or paperboard with:
        1. the answer to someone else’s problem (written above)
        2. their own problem.
      • The student that has “12″ as the written answer above their problem will start the game.
      • This student would read their problem aloud.
      • If the classmate that had the correct answer to this problem on their sheet of paper stands and reads their answer aloud, then they receive the prize.
      • This student then reads the problem on their page, and the mathematical treasure hunt continues.
      • The play would continue until all of the problems and answers were stated.
      • The difficult problems could have a higher prize compared to the easy ones.
      • Sample answers:
        1, 3, 9, 27, 81, 64, 11, 4, 54, 85, 5, 16, 47, 15, 2, 12
    2. Review of past lesson
      • By now, you probably have a clear idea of what a sequence is since we discussed it during our last class period. But can you differentiate it from a series?
            5, 10, 15, 20, 25 is a finite

        sequence

            5 + 10 + 15 + 20 + 25 is a finite

        series

            1, 3, 5, 7, 9… is an infinite

        sequence

            1 + 3 + 5 + 9 + … is an infinite

        series

      • From the preceding examples, we can see that an arithmetic sequence has a common difference between terms. The arithmetic series is an indicated sum of the terms of an arithmetic sequence. In other words: an arithmetic series is the indicated sum of an arithmetic sequence .
      • The sum of the finite series 5 + 10 + 15 + 20 + 25 is equal to 75. Adding the terms of the finite sequence may not require much work when the number of terms to be added is small. However, when many terms are to be added, a lot more time and effort are needed.
  • Lesson proper:
    • We have here a picture of a man lying under a tree waiting for a guava fruit to fall down. His name is Juan, Juan Tamad. When his friend Pedro walked by and saw Juan. He said, “Juan, don’t wait for the guava to fall down. The fruit might not fall down until it is already rotten and there might even be a worm in it. Why don’t you make some stair steps to climb in order to reach the guava. The guava fruit is in a 25 foot tree and the bricks you have to make the steps out of are 1 foot bricks. How many bricks would Juan need to use in making his stairs? How many bricks should Juan use to reach the guava fruit ?

      Solution:

      1, 2, 3, 4, 5, 6…, 25 =

      (1 + 25) + (2 + 24) + (3 + 23) + (4 + 22) + (5 + 21) + (6 + 20) + (7 + 19) + (8 + 18) + (9 + 17) + (10 + 16) + (11 + 15) + (12 + 14) + 13 =

      Total bricks = (1 + 25 )( 25 )/2 = 325 bricks

    • Did you know that the German mathematician, Carl Friedrich Gauss, was also called the Prince of Mathematics? In school, when his teacher assigned the problem of summing the integers from 1 up to 100 to keep his students busy, Carl immediately wrote the correct answer, 5050, on his slate. Carl drew a figure like this:

      1, 2, 3, 4, 5, 6, 7, 8, 9, 10…, 50, 51…, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100

      101

      101

      101

      101

      101

      101

      101

      101

      101

      101

      101

      101

      101

      Sum of numbers from 1 up to 100 = (1 + 100)(100)/2

      = 101(100)/2

      = 10100/2

      = 5050

    • Last February, Sharon Cuneta held a mega-concert at the coliseum. How many chairs are in the coliseum if there are 40 rows and the first row contains 200 seats and the second row contains 250 seats and the succeeding rows follow this arithmetic sequence?

      Solution:

      First we will need to find the last term using the explicit formula

      (a 1 ) + (n – 1)d

      to find a particular term n (the last one) in a sequence. Remember, the first term is a 1 , the common difference is d, and the number of terms is n.

      200 + 250 + 300 + … + [(a 1 ) + (n – 1)d]

      a 1 = 200, a 2 = 250, d difference between rows = a 2 – a 1 = 50, n rows = 40

      200 + 250 + 300 + … + [200 + (40 - 1)50]
      200 + 250 + 300 + … + [200 + (39)(50)]
      200 + 250 + 300 + … + [200 + 1950]
      200 + 250 + 300 + … + 2150

      Then we can apply the formula we discovered above:

      (a 1 + a n )(n/2) = (200 + 2150)(40/2) = 47,000

      Other ways of stating the same formula are:

      (a 1 + a n )/2 * n = (200 + 2150)/2 * 40 = 2350/2 * 40 = 1175 * 40 = 47,000

      n(a 1 + a n )/2 = 40(200 + 2150)/2 = 40(2350)/2 = 94000/2 = 47,000

  • Generalization :
    • Finding the sum of a small finite arithmetic sequence with small terms is easy. Just jot down the terms and add.
    • To find the sum of a longer sequence, you need to memorize the formula and know three things: the first term, the last term and the number of terms.
    • One must first have mastered the arithmetic sequence formula for finding an unknown given term (last term) before an individual can acquire the knowledge to find the sum of a given longer sequence.

      a n = a 1 + (n – 1)d

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