Challenging +1.8 This question requires understanding of recursive sequences, logarithms for exponential growth, infinite products, and convergence conditions. Part (a) is routine logarithmic manipulation, but parts (b) and (c) demand sophisticated analysis: deriving the general term of v_n, finding convergence conditions for an infinite product (requiring sum of exponents to converge), and recognizing the pattern in nested radicals. The combination of these non-standard techniques and the need for exact algebraic manipulation places this well above average difficulty, though it's structured enough to guide students through the reasoning.
4. A sequence is defined by \(u _ { 1 } = 3 , u _ { n + 1 } = u _ { n } ^ { r }\) for \(n \geq 1\).
a) In the case where \(r = \frac { 6 } { 5 }\) find the smallest value of \(n\) such that \(u _ { n } > 10 ^ { 50 }\).
A convergent sequence is defined by \(v _ { 1 } = u _ { 1 } , v _ { n + 1 } = u _ { n + 1 } v _ { n }\) for \(n \geq 1\).
b) Given that the limit of this sequence is greater than 100 , find the range of possible values of \(r\), giving your answer in exact form.
c) Evaluate the infinite product:
$$2 \times \sqrt [ 3 ] { 4 } \times \sqrt [ 3 ] { \sqrt [ 3 ] { 16 } } \times \sqrt [ 3 ] { \sqrt [ 3 ] { \sqrt [ 3 ] { 256 } } } \cdots$$
[Question 4 - Continued] [0pt]
[Question 4 - Continued] [0pt]
[Question 4 - Continued]
4. A sequence is defined by $u _ { 1 } = 3 , u _ { n + 1 } = u _ { n } ^ { r }$ for $n \geq 1$.\\
a) In the case where $r = \frac { 6 } { 5 }$ find the smallest value of $n$ such that $u _ { n } > 10 ^ { 50 }$.
A convergent sequence is defined by $v _ { 1 } = u _ { 1 } , v _ { n + 1 } = u _ { n + 1 } v _ { n }$ for $n \geq 1$.\\
b) Given that the limit of this sequence is greater than 100 , find the range of possible values of $r$, giving your answer in exact form.\\
c) Evaluate the infinite product:
$$2 \times \sqrt [ 3 ] { 4 } \times \sqrt [ 3 ] { \sqrt [ 3 ] { 16 } } \times \sqrt [ 3 ] { \sqrt [ 3 ] { \sqrt [ 3 ] { 256 } } } \cdots$$
[Question 4 - Continued]\\[0pt]
[Question 4 - Continued]\\[0pt]
[Question 4 - Continued]\\
\hfill \mbox{\textit{SPS SPS FM 2022 Q4 [20]}}