Solve: `2/(sqrt1 + sqrt2) + 2/(sqrt2 + sqrt3) + 2/(sqrt3 + sqrt4) + 2/(sqrt4 + sqrt5) + ... + 2/(sqrt8 + sqrt9)`

justaguide | College Teacher | (Level 2) Distinguished Educator

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The value of `2/(sqrt1 + sqrt2) + 2/(sqrt2 + sqrt3) + 2/(sqrt3 + sqrt4) + 2/(sqrt4 + sqrt5) + ... + 2/(sqrt8 + sqrt9)` has to be determined.

`2/(sqrt1 + sqrt2) + 2/(sqrt2 + sqrt3) + 2/(sqrt3 + sqrt4) + 2/(sqrt4 + sqrt5) + ... + 2/(sqrt8 + sqrt9) `

=> `2(1/(sqrt1 + sqrt2) + 1/(sqrt2 + sqrt3) + 1/(sqrt3 + sqrt4) + 1/(sqrt4 + sqrt5) + ... + 1/(sqrt8 + sqrt9))`

=> `2((sqrt1 - sqrt2)/((sqrt1 + sqrt2)(sqrt1 - sqrt2)) + (sqrt2 + sqrt3)/((sqrt2 - sqrt3)(sqrt2 + sqrt3)) `

`+ (sqrt3 - sqrt4)/((sqrt3 + sqrt4)(sqrt3 - sqrt4)) + (sqrt4 - sqrt5)/((sqrt4 - sqrt5)(sqrt4 + sqrt5)) + ... + (sqrt8 - sqrt9)/((sqrt8 - sqrt9)(sqrt8 + sqrt9)))`

=> `2((sqrt1 - sqrt2)/(1 - 2) + (sqrt2 - sqrt3)/(2 - 3) + (sqrt3 - sqrt4)/(3 - 4) + (sqrt4 - sqrt5)/(4 - 5) + ... + (sqrt8 - sqrt9)/(8 - 9))`

=> `-2*(sqrt1 - sqrt2 + sqrt2 - sqrt3 + sqrt3 - sqrt4 + sqrt4 - sqrt5 + ... + sqrt8 - sqrt9)`

=> `-2*(sqrt1 - sqrt 9)`

=> `-2*(1 - 3)`

=> `-2*-2`

=> `4`

The value of `2/(sqrt1 + sqrt2) + 2/(sqrt2 + sqrt3) + 2/(sqrt3 + sqrt4) + 2/(sqrt4 + sqrt5) + ... + 2/(sqrt8 + sqrt9) = 4`