1/1*2 +1/2*3+...+1/n(n+1)= n/n+1 proof by mathematical induction
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Base case (n = 1):

• On the left side: 1/(1×2) = 1/2

• On the right side: 1/(1 + 1) = 1/2

Induction hypothesis: Assume the statement is true for n = k ; that is,

1/(1×2) + 1/(2×3) + … + 1/(k × (k + 1))) = k/(k + 1)

Inductive step (n = k + 1):

1/(1×2) + 1/(2×3) + … + 1/(k × (k + 1))) + 1/((k + 1) × (k + 2)))

= k/(k + 1) + 1/((k + 1) × (k + 2))

= (k × (k + 2) + 1) / ((k + 1) × (k + 2))

= (k ² + 2k + 1) / ((k + 1) × (k + 2))

= (k + 1)² / ((k + 1) × (k + 2))

= (k + 1) / (k + 2)

and this is what we wanted to show.

a

step-by-step explanation:

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