Prove summation with logarithms

A question is this type if and only if it asks to prove by induction a summation formula involving logarithmic terms (e.g., ∑log(2r-1), ∑r·ln((r+1)/r)).

2 questions · Challenging +1.0

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Edexcel F1 2023 January Q9

6 marks Challenging +1.2

Prove by induction that for all positive integers \(n\)

$$\sum _ { r = 1 } ^ { n } \log ( 2 r - 1 ) = \log \left( \frac { ( 2 n ) ! } { 2 ^ { n } n ! } \right)$$

CAIE FP1 2017 June Q3

6 marks Standard +0.8

3 Prove, by mathematical induction, that \(\sum _ { r = 1 } ^ { n } r \ln \left( \frac { r + 1 } { r } \right) = \ln \left( \frac { ( n + 1 ) ^ { n } } { n ! } \right)\) for all positive integers \(n\). [6]