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A piece of string of length 5m long is cut into n pieces in such a way that the lengths of the pieces are an arithmetic sequence. If the lengths of the longest and the shortest pieces are 1m and 25cm respectively, calculate n.

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  1. In an arithmetic progression:

    an = a + ( n - 1 ) d

    where

    a = the initial term

    d = the common difference of successive members

    an = the nth term

    Lengths shortest pieces:

    a1 = a + ( 1 - 1 ) d = a + 0 ∙ d = a

    a1 = a = 25 cm

    Lengths longest pieces:

    an = 1 m = 100 cm

    an = a + ( n - 1 ) d

    100 = 25 + ( n - 1 ) d

    The sum of n terms of an arithmetic progression:

    Sn = ( n / 2 ) [ 2 a + ( n -1 ) d ]

    In this case a = 25 cm so:

    Sn = ( n / 2 ) [ 2 ∙ 25 + ( n -1 ) d ]

    Sn = ( n / 2 ) [ 50 + ( n -1 ) d ]

    The sum of n terms of this arithmetic progression is 5 m

    Sn = 5 m = 500 cm

    500 = ( n / 2 ) [ 50 + ( n -1 ) d ]

    Now you must solve system of two equations:

    25 + ( n - 1 ) d = 100

    ( n / 2 ) [ 50 + ( n -1 ) d ] = 500

    The solution is:

    d = 75 / 7 , n = 8

    Your arithmetic progression:

    a1 = 25

    a2 = 25 + 75 / 7 = 175 / 7 + 75 / 7 = 250 / 7

    a3 = 250 / 7 + 75 / 7 = 325 / 7

    a4 = 325 / 7 + 75 / 7 = 400 / 7

    a5 = 400 / 7 + 75 / 7 = 475 / 7

    a6 = 475 / 7 + 75 / 7 = 550 / 7

    a7 = 550 / 7 + 75 / 7 = 625 / 7

    a8 = 625 / 7 + 75 / 7 = 700 / 7 = 100

    You can check the sum of this arithmetic progression.

    a1 + a2 + a3 + a4 + a5 + a6 + a7 =

    25 + 250 / 7 + 325 / 7 + 400 / 7 + 475 / 7 + 550 / 7 + 625 / 7 + 100 =

    25 + ( 250 / 7 + 325 / 7 + 400 / 7 + 475 / 7 + 550 / 7 + 625 / 7 ) + 100 =

    25 + 2625 / 7 + 100 = 25 + 375 + 100 = 500

    The sum of this arithmetic progression = 500 cm

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  2. My little typo.

    the sum of this arithmetic progression is:

    a1 + a2 + a3 + a4 + a5 + a6 + a7 + a8 =

    25 + 250 / 7 + 325 / 7 + 400 / 7 + 475 / 7 + 550 / 7 + 625 / 7 + 100

    ...

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