Lesotho General Certificate of Secondary Education: Mathematics: Paper 3

$ (a.) \\[3ex] 2, ..., 18, 54, 162, ..., ... \\[3ex] \dfrac{54}{18} = \dfrac{162}{54} = 3 \quad\text{(common ratio)} \\[5ex] \text{This is a Geometric Sequence} \\[3ex] \text{2nd term} = 2 \times 3 = 6 \\[3ex] \text{6th term} = 162 \times 3 = 486 \\[5ex] (b.) \\[3ex] -1, -1, 1, 5, 11, ... \\[3ex] $ A quick observation of the sequence shows that it is not a/an:
Arithmetic Sequence because we do not have a common difference
Geometric Sequence because we do not have a common ratio
So, let us try to see if it is a Quadratic Sequence

$ \underline{\text{1st Difference Between Consecutive Terms}} \\[3ex] -1 - (-1) = -1 + 1 = 0 \\[3ex] 1 - (-1) = 1 + 1 = 2 \\[3ex] 5 - 1 = 4 \\[3ex] 11 - 5 = 6 \\[5ex] \text{Sequence of 1st Differences: } 0, 2, 4, 6 \\[3ex] \underline{\text{2nd Difference Between Consecutive Terms of the Sequence of First Differences}} \\[3ex] 2 - 0 = 2 \\[3ex] 4 - 2 = 2 \\[3ex] 6 - 4 = 2 \\[3ex] \text{2nd Differences are the same: The initial sequence is a Quadratic Sequence} \\[3ex] \text{General Form} = n\text{th term: } y = ax^2 + bx + c \\[3ex] \text{where: } \\[3ex] y = \text{outputs of terms} \\[3ex] x = \text{input of terms} \\[3ex] a, b, c = \text{coefficients and constant in the }n\text{th term} \\[3ex] \text{Initial Sequence: } y = -1, -1, 1, 5, 11, ... \\[3ex] ax^2 + bx + c = y \\[3ex] \text{When } x = 1 \quad\text{1st term: } y = -1 \\[3ex] a(1)^2 + b(1) + c = -1 \\[3ex] a + b + c = -1 \quad eqn.(1) \\[5ex] \text{When } x = 2 \quad\text{2nd term: } y = -1 \\[3ex] a(2)^2 + b(2) + c = -1 \\[3ex] 4a + 2b + c = -1 \quad eqn.(2) \\[5ex] \text{When } x = 3 \quad\text{2nd term: } y = 1 \\[3ex] a(3)^2 + b(3) + c = 1 \\[3ex] 9a + 3b + c = 1 \quad eqn.(3) \\[5ex] eqn.(2) - eqn.(1) \implies \\[3ex] 3a + b = 0 \quad eqn.(4) \\[5ex] eqn.(3) - eqn.(2) \implies \\[3ex] 5a + b = 2 \quad eqn.(5) \\[5ex] eqn.(5) - eqn.(4) \implies \\[3ex] 2a = 2 \\[3ex] a = \dfrac{2}{2} \\[5ex] a = 1 \\[5ex] \text{Substitute for } a \text{ in } eqn.(4) \\[3ex] 3(1) + b = 0 \\[3ex] 3 + b = 0 \\[3ex] b = 0 - 3 \\[3ex] b = -3 \\[5ex] \text{Substitute for } a \text{ and } b \text{ in } eqn.(1) \\[3ex] 1 + (-3) + c = -1 \\[3ex] -2 + c = -1 \\[3ex] c = -1 + 2 \\[3ex] c = 1 \\[5ex] n\text{th term, } y = x^2 - 3x + 1 \\[5ex] (c.) \\[3ex] a = \text{1st term} = 4 \\[3ex] d = \text{common difference} \\[3ex] n = \text{number of terms} \\[3ex] T_n = n\text{th term} \\[5ex] (i.) \\[3ex] T_n = a + d(n - 1) \\[3ex] T_5 = 4 + d(5 - 1) = 16 \\[3ex] 4 + 4d = 16 \\[3ex] 4d = 16 - 4 \\[3ex] 4d = 12 \\[3ex] d = \dfrac{12}{4} \\[5ex] d = 3 \\[5ex] (ii.) \\[3ex] T_n = 4 + 3(n - 1) \\[3ex] = 4 + 3n - 3 \\[3ex] = 3n + 1 $

$ (a.) \\[3ex] \text{Teboho : Sebolelo} = \text{Ratio of } 3 : 5 \\[3ex] \text{Sum of ratios} = 3 + 5 = 8 \\[3ex] \text{Amount shared} = M72 000 \\[5ex] (i.) \\[3ex] \text{Teboho's share} = \dfrac{3}{8} \times 72000 \\[5ex] = 3 \times 900 \\[3ex] = M2700 \\[5ex] (ii.) \\[3ex] \text{Percentage of Sebolelo's share} = \dfrac{5}{8} \times 100\% \\[5ex] = 62.5\% \\[5ex] (b.) \\[3ex] SI = P \times r \times t \\[3ex] \text{Where} \\[3ex] SI = \text{Simple Interest} \\[3ex] P = \text{Principal} \\[3ex] r = \text{rate} \\[3ex] t = \text{time} = 3\text{ years} \\[3ex] $ Let us represent the information in a table for easier computation.
Let the:
part of M10 000 invested at 2.5% per annum simple interest = $x$
part invested at 1.9% per annum simple interest = $10000 - x$

Account	Principal, P (M)	rate, r (%)	time, t (years)	Simple Interest, SI (M)
Part 1	$x$	$ 2.5\% = \dfrac{2.5}{100} = 0.025 $	$3$	$0.075x$
Part 2	$10000 - x$	$ 1.9\% = \dfrac{1.9}{100} = 0.019 $	$3$	$0.057(10000 - x)$

$ 0.075x + 0.057(10000 - x) = 678 \quad\text{(total interest of M678)} \\[3ex] 0.075x + 570 - 0.057x = 678 \\[3ex] 0.018x = 678 - 570 \\[3ex] 0.018x = 108 \\[3ex] x = \dfrac{108}{0.018} \\[5ex] x = M6000 \\[3ex] $ The amount invested at 2.5% is M6000