2022 · Q8
What is the value of the following?(1)/(5√4 + 4√5) + (1)/(6√5 + 5√6) + (1)/(7√6 + 6√7) + (1)/(8√7 + 7√8) + (1)/(9√8 + 8√9)
Elementary MathematicsAlgebra
- A1/√6
- B1/2
- C1
- D1/6
Explanation
Rationalise each term: 1/(5√4+4√5) = (√5-√4)/(√20x1) = 1/√4-1/√5 = 1/2-1/√5, and similarly each term equals 1/√n-1/√(n+1). The series telescopes: (1/2-1/√5)+(1/√5-1/√6)+...+(1/√8-1/3) = 1/2-1/3 = 1/6. Hence the keyed option is D.