Funatic Maths

Sequences-and-series

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    - Paper 1

    Question 2

    2.12.1 7 ;x ;y ;−11 ;...7 \,; x \,; y \,; -11 \,; ... is an arithmetic sequence. Determine the values of xx and yy. (4)(4)
    2.22.2 Given the quadratic number pattern:  −3 ;6 ;27 ;60 ;...\, -3 \,; 6 \,; 27 \,; 60 \,; ...
    2.2.1\quad 2.2.1 Determine the general term of the pattern in the form  Tn=an2+bn+c\:T_n=an^2+bn+c. (4)(4)
    2.2.2\quad 2.2.2 Calculate the value of the 50th50^{th} term of the pattern. (2)(2)
    2.2.3\quad 2.2.3 Show that the sum of the first nn first-differences of this pattern can be given by  Sn=6n2+3n\:S_n=6n^2+3n . (3)(3)
    2.2.4\quad 2.2.4 How many consecutive first-differences were added to the first term of the quadratic number pattern to obtain a term in the quadratic number pattern that has a value of 21 06021\,060? (4)(4)
    [17]\textbf{[17]}

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    - Paper 1

    Question 2

    2.12.1 Given the quadratic sequence:  321 ;290 ;261 ;234 ;...\:321 \,; 290 \,; 261 \,; 234 \,; ...
    2.1.1\quad 2.1.1 Write down the values of the next TWO terms of the sequence. (2)(2)
    2.1.2\quad 2.1.2 Determine the general term of the sequence in the form  Tn=an2+bn+c\:T_n=an^2+bn+c. (4)(4)
    2.1.3\quad 2.1.3 Which term(s) of the sequence will have a value of 7474? (4)(4)
    2.1.4\quad 2.1.4 Which term in the sequence has the least value? (2)(2)
    2.22.2 Given the geometric series:  58+516+532+...=K\, \frac{\displaystyle5}{\displaystyle8}+\frac{\displaystyle5}{\displaystyle16}+\frac{\displaystyle5}{\displaystyle32}+...=K
    2.2.1\quad 2.2.1 Determine the value of KK if the series has 2121 terms. (3)(3)
    2.2.2\quad 2.2.2 Determine the largest value of nn for which  Tn>58192\:T_n>\frac{\displaystyle5}{\displaystyle8192} (4)(4)
    [19]\textbf{[19]}

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    Question 3

    3.13.1 Prove that  ∑k=1∞4.32−k\: \sum\limits_{k=1}^{\infty} 4.3^{2-k} is a convergent geometric series. Show ALL your calculations. (3)(3)
    3.23.2 If  ∑k=p∞4.32−k=29\: \sum\limits_{k=p}^{\infty} 4.3^{2-k}=\frac{\displaystyle 2}{\displaystyle 9}, determine the value of pp. (5)(5)
    [8]\textbf{[8]}

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    - Paper 1

    Question 3

    3.13.1 Without using a calculator, determine the value of:  ∑y=3101y−2−∑y=3101y−1\: \sum\limits_{y=3}^{10}\frac{\displaystyle 1}{\displaystyle y-2} - \sum\limits_{y=3}^{10}\frac{\displaystyle 1}{\displaystyle y-1} (3)(3)
    3.23.2 A steel pavilion at a sports ground comprises of a series of 1212 steps, of which the first 33 are shown in the diagram below.
    Each step is 5 m5\,m wide. Each step has a rise of 13 m\frac{\displaystyle 1}{\displaystyle 3}\,m and has a tread of 23 m\frac{\displaystyle 2}{\displaystyle 3}\,m, as shown in the diagram below.
    Image
    The open side (shaded on sketch) on each side of the pavilion must be covered with metal sheeting. Calculate the area (in m2m^2) of metal sheeting needed to cover both open sides. (6)(6)
    [9]\textbf{[9]}
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