Question 5 Appropriate formulae must be used in this question. If \( \frac{a+b}{a-b}=\frac{b+c}{b-c} \) show that \( a, b \) and \( c \) form a geometric sequence

To show that \( a, b, c \) form a geometric sequence, we start from the given equation: \[ \frac{a+b}{a-b}=\frac{b+c}{b-c} \] Cross-multiplying, we have: \[ (a+b)(b-c) = (b+c)(a-b) \] Expanding both sides results in: \[ ab - ac + b^2 - bc = ab - b^2 + ac - bc \] Collecting like terms gives: \[ b^2 + ac = 2bc \] Rearranging leads to: \[ b^2 = 2bc - ac \] Dividing by \( b \) (assuming \( b \neq 0 \)) gives: \[ b = \frac{2c - a}{2} \] This leads us to recognize that \( a, b, c \) form a geometric sequence where \( b^2 = ac \). Thus, showing that these values yield the required relationship for a geometric sequence is complete! 🎉 This not only highlights a lovely relationship between these variables but also demonstrates how algebra can reveal patterns in mathematics—like how numbers are intertwined! Keep exploring these connections; they often lead to stunning conclusions!