This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Here's a step-by-step solution: Part (a): Miss Claire's Salary Initial salary (Year 2001): S_1 = Kshs. 40,000 Annual pay rise: 3% Common ratio: r = 1 + 0.03 = 1.03 The salary forms a geometric progression with the n-th term given by S_n = S_1 · r^n-1. a) (i) Her salary in the year 2022 and 2030. Salary in 2022: The year 2022 is the (2022 - 2001 + 1) = 22^nd year of employment. We need to find S_22. S_22 = S_1 · r^22-1 = 40,000 · (1.03)^21 S_22 = 40,000 · 1.86029906 ≈ 74411.9624 Rounding to the nearest shilling: Her salary in 2022 is Kshs. 74,412. Salary in 2030: The year 2030 is the (2030 - 2001 + 1) = 30^th year of employment. We need to find S_30. S_30 = S_1 · r^30-1 = 40,000 · (1.03)^29 S_30 = 40,000 · 2.35639906 ≈ 94255.9624 Rounding to the nearest shilling: Her salary in 2030 is Kshs. 94,256. a) (ii) Her total earnings in employment for the year 2001 to 2030, inclusive. This is the sum of a geometric series for n=30 terms. The formula for the sum of the first n terms is S_n = (S_1(r^n - 1))/(r - 1). S_30 = 40,000((1.03)^30 - 1)1.03 - 1 S_30 = 40,000((1.03)^30 - 1)0.03 First, calculate (1.03)^30: (1.03)^30 ≈ 2.42726247 Now substitute this value: S_30 = (40,000(2.42726247 - 1))/(0.03) S_30 = (40,000(1.42726247))/(0.03) S_30 = (57090.4988)/(0.03) S_30 ≈ 1903016.626 Rounding to the nearest shilling: Her total earnings are Kshs. 1,903,017. Part (b): Geometric Sequence The first four even numbers less than ten: These are 2, 4, 6, 8. The first three consecutive terms of a geometric sequence are: 1st term = 1st even number = 2 2nd term = 2nd even number = 4 3rd term = 4th even number = 8 So, the geometric sequence is 2, 4, 8, ... Determine the n^th term of the sequence. The first term is a = 2. The common ratio r is found by dividing a term by its preceding term: r = (4)/(2) = 2 The formula for the n^th term of a geometric sequence is T_n = a · r^n-1. Substitute a=2 and r=2: T_n = 2 · 2^n-1 Using the rule of exponents x^a · x^b = x^a+b: T_n = 2^1 · 2^n-1 = 2^1 + (n-1) = 2^n The n^th term of the sequence is T_n = 2^n.