| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2023 |
| Session | June |
| Marks | 6 |
| Topic | Arithmetic Sequences and Series |
| Type | Logarithmic arithmetic progression |
| Difficulty | Moderate -0.8 This question tests basic logarithm laws and arithmetic series summation in a straightforward manner. Part (a) requires recognizing that ln(p^n) = n·ln(p) and summing 1+2+...+11 = 66. Part (b) uses similar techniques with ln(8) = 3ln(2). Part (c) is a simple inequality ln(2p²) < 0 leading to p < 1/√2. All steps are routine applications of standard A-level techniques with no problem-solving insight required, making it easier than average but not trivial due to the algebraic manipulation involved. |
| Spec | 1.04g Sigma notation: for sums of series1.06f Laws of logarithms: addition, subtraction, power rules |
Given that $p$ is a positive constant,
(a) show that
$$\sum_{n=1}^{11} \ln(p^n) = k \ln p$$
where $k$ is a constant to be found, [2]
(b) show that
$$\sum_{n=1}^{11} \ln(8p^n) = 33\ln(2p^2)$$ [2]
(c) Hence find the set of values of $p$ for which
$$\sum_{n=1}^{11} \ln(8p^n) < 0$$
giving your answer in set notation. [2]
\hfill \mbox{\textit{SPS SPS SM Pure 2023 Q18 [6]}}