1.04j Sum to infinity: convergent geometric series |r|<1

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SPS SPS FM 2025 October Q11
8 marks Standard +0.3
The functions f and g are defined by $$\text{f}(x) = \frac{3}{2}\ln x \quad x > 0$$ $$\text{g}(x) = \frac{4x + 3}{2x + 1} \quad x > 0$$
  1. Find gf(\(\text{e}^2\)) writing your answer in simplest form. [2]
  2. Find the range of the function fg. [2]
  3. Given that f(8) and f(2) are the second and third terms respectively of a geometric series, find the sum to infinity of this series, giving your answer in the form \(a \ln 2\) where \(a\) is rational. [4]
SPS SPS SM 2025 October Q4
6 marks Moderate -0.8
  1. A sequence has terms \(u_1, u_2, u_3, \ldots\) defined by \(u_1 = 3\) and \(u_{n+1} = u_n^2 - 5\) for \(n \geq 1\).
    1. Find the values of \(u_2\), \(u_3\) and \(u_4\). [2]
    2. Describe the behaviour of the sequence. [1]
  2. The second, third and fourth terms of a geometric progression are 12, 8 and \(\frac{16}{3}\). Determine the sum to infinity of this geometric progression. [3]
OCR H240/03 2018 March Q2
8 marks Moderate -0.3
The first term of a geometric progression is 12 and the second term is 9.
  1. Find the fifth term. [3]
The sum of the first \(N\) terms is denoted by \(S_N\) and the sum to infinity is denoted by \(S_\infty\). It is given that the difference between \(S_\infty\) and \(S_N\) is at most 0.0096.
  1. Show that \(\left(\frac{3}{4}\right)^N \leqslant 0.0002\). [3]
  2. Use logarithms to find the smallest possible value of \(N\). [2]
OCR H240/01 2017 Specimen Q7
10 marks Moderate -0.8
Business A made a £5000 profit during its first year. In each subsequent year, the profit increased by £1500 so that the profit was £6500 during the second year, £8000 during the third year and so on. Business B made a £5000 profit during its first year. In each subsequent year, the profit was 90% of the previous year's profit.
  1. Find an expression for the total profit made by business A during the first \(n\) years. Give your answer in its simplest form. [2]
  2. Find an expression for the total profit made by business B during the first \(n\) years. Give your answer in its simplest form. [3]
  3. Find how many years it will take for the total profit of business A to reach £385 000. [3]
  4. Comment on the profits made by each business in the long term. [2]
Pre-U Pre-U 9794/1 2011 June Q11
9 marks Standard +0.3
An arithmetic progression has first term \(a\) and common difference \(d\). The first, ninth and fourteenth terms are, respectively, the first three terms of a geometric progression with common ratio \(r\), where \(r \neq 1\).
  1. Find \(d\) in terms of \(a\) and show that \(r = \frac{5}{3}\). [7]
  2. Find the sum to infinity of the geometric progression in terms of \(a\). [2]