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Math Help - Arithmetic series..... Log.. other problems..

  1. #1
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    Arithmetic series..... Log.. other problems..

    An arithmetic series has five term a and common difference d.
    The sum of the first 31 terms of the series is 310.


    a) Show that a + 15d = 10

    > S31 = 31/2 ( 2a + (31-1) d)
    > 31( a + 15d ) = 310
    > a + 15d = 310/31; a + 15d = 10


    b)Given also that the 21st term is twise the 16th term, find the value of d.

    ???

    c)the nth term of the series is un. Given that
    K
    Sum un = 0, find the value of K.
    n = 1

    ???

    ----------------------------------
    Find the value of x in each of the following:

    (A). log9x = 0
    (B). log9x = 1/2

    of coz i noe how to do it on the calculator with the answers 1 for (A) and 3 for (B), but how do i get it withour a calculator??

    ----------------------------------

    A curve has equation y = 1/x^3 + 48x.

    a) Find dy/dx.

    >> dy/dx = 3x^-4 + 48

    b) Hence find the equation of each of the two tangents to the curve that are parallel to the x-axis.

    >> -3x^-4 + 48 = 0
    >> x^-4 = 16
    >> x = +-1/2


    c) Find the equation of the normal to the curve at the point (1,49)

    ???????????
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  2. #2
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    Quote Originally Posted by ansonbound View Post
    An arithmetic series has five term a and common difference d.
    The sum of the first 31 terms of the series is 310.


    a) Show that a + 15d = 10

    > S31 = 31/2 ( 2a + (31-1) d)
    > 31( a + 15d ) = 310
    > a + 15d = 310/31; a + 15d = 10


    b)Given also that the 21st term is twise the 16th term, find the value of d.

    ??? \color{blue}a+15d=10......clue1

    \color{blue}a+(21-1)d=2[a+(16-1)d].......clue2

    The clues allow you to solve the simultaneous equations for "d"

    \color{blue}a+20d=2a+30d

    subtract the left from the right

    \color{blue}0=a+10d

    use your first result....clue1

    \color{blue}a+15d-(a+10d)=10-0

    \color{blue}5d=10\ \Rightarrow\ d=2,\ a=-20




    c)the nth term of the series is un. Given that
    K
    Sum un = 0, find the value of K.
    n = 1

    ??? \color{blue}\frac{k}{2}[2a+(k-1)d]=0

    \color{blue}\frac{k}{2}[-40+(k-1)2]=0

    Inside the brackets is zero, so

    \color{blue}(k-1)2=40\ \Rightarrow\ k-1=20\ \Rightarrow\ k=21


    ----------------------------------
    Find the value of x in each of the following:

    (A). log9x = 0
    (B). log9x = 1/2

    of coz i noe how to do it on the calculator with the answers 1 for (A) and 3 for (B), but how do i get it withour a calculator??

    \color{blue}log_9x=0\ \Rightarrow\ x=9^0=\frac{9^y}{9^y}=1

    \color{blue}log_9x=\frac{1}{2}\ \Rightarrow\ x=9^{\frac{1}{2}}=\sqrt{9}=3

    ----------------------------------

    A curve has equation y = 1/x^3 + 48x.

    a) Find dy/dx.

    >> dy/dx = -3x^-4 + 48

    b) Hence find the equation of each of the two tangents to the curve that are parallel to the x-axis.

    >> -3x^-4 + 48 = 0
    >> x^-4 = 16
    >> x = +-1/2


    c) Find the equation of the normal to the curve at the point (1,49)

    ???????????

    The slope of the tangent at x=1 is

    \color{blue}-3(1)^{-4}+48=45

    The normal is perpendicular to the tangent, hence invert and negate the tangent slope


    The slope of the normal is -1/45 and the normal contains the point (1,49)

    Hence, the normal equation at that point is

    \color{blue}y-49=-\frac{1}{45}(x-1)
    .
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  3. #3
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    Thankyou very much!!! now i understand!!!!
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