Maths

posted by Anonymous

The 5th term of an arithmetic progression is 3times of 2nd term and 12th term exceeds 2times of 6th term by 1. Find the 16th term

  1. Bosnian

    a5 = 3 a2

    a12 = 2 a6 + 1


    For AP:

    an = a1 + ( n - 1 ) d

    a1 = initial term of an arithmetic progression

    d = common difference of successive members


    a2 = a1 + ( 2 - 1 ) d

    a2 = a1 + 1 * d

    a2 = a1 + d


    a5 = a1 + ( 5 - 1 ) d

    a5 = a1 + 4 d


    a6 = a1 + ( 6 - 1 ) d

    a6 = a1 + 5 d


    a12 = a1 + ( 12 - 1 ) d

    a12 = a1 + 11 d


    Now:

    a5 = 3 a2

    a5 = 3 ( a1 + d )

    a5 = 3 a1 + 3 d


    a5 = a5

    a1 + 4 d = 3 a1 + 3 d Subtract a1 to both sides

    a1 + 4 d - a1 = 3 a1 + 3 d - a1

    4 d = 2 a1 + 3 d Subtract 3 d to both sides

    4 d - 3 d= 2 a1 + 3 d - 3 d

    d = 2 a1


    a12 = 2 a6 + 1

    a1 + 11 d = 2 ( a1 + 5 d ) + 1

    a1 + 11 d = 2 a1 + 10 d + 1 Subtract a1 to both sides

    a1 + 11 d - a1 = 2 a1 + 10 d + 1 - a1

    11 d = a1 + 10 d + 1 Subtract 10 d to both sides

    11 d - 10 d = a1 + 10 d + 1 - 10 d

    d = a1 + 1


    d = d

    2 a1 = a1 + 1 Subtract 1 to both sides

    2 a1 - 1 = a1 + 1 - 1

    2a1 - 1 = a1 Subtract a1 to both sides

    2a1 - 1 - a1 = a1 - a1

    a1 - 1 = 0 Add 1 to both sides

    a1 - 1 + 1 = 0 + 1

    a1 = 1


    d = 2 a1

    d = 2 * 1

    d = 2


    an = a1 + ( n - 1 ) d

    a16 = a1 + ( 16 - 1 ) d

    a16 = a1 + 15 d

    a16 = 1 + 15 * 2

    a16 = 1 + 30

    a16 = 31

  2. Steve

    a+4d = 3(a+d)
    a+11d = 2(a+5d)+1

    Solve for a and d, then

    T16 = a+15d

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