Question

Use mathematical induction to prove that:
1/1*2+1/2*3+1/3*4+...+1/n*(n+1)=n/n+1

Have no idea where to start!

Answer 1

well... one would think at the base case :P

Answer 2

so its obviously having to prove it for n>1

Answer 3

n>=1

Answer 4

so check n=1 , its true , thats the first step done

Answer 5

then assume it is true for n=k , ie that S(k) { the sum of k terms } = k / (k+1)

Answer 6

now we need to prove it is true for n=k+1 , ie that S (k+1) = (k+1)/ ( k+2)

where we got the RHS , by substituting n=k+1 in the summation

Answer 7

now there is a result that S(k+1) = S(k) + T(k+1)

that is , in words, "the sum of k+1 terms is equal to sum of the first k terms , plus the (k+1)th term "

Answer 8

like saying, sum 1 + 2 + 3 + 4 , well thats equal to summing the first 3 ( to get 6 ) and then adding on 4 ( the (k+1)th term )

Answer 9

now , back to the question ,

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

Answer 10

S(k+1) = [ k/(k+1) ] + 1/ [ (k+1)(k+2)]

where the first fraction came from our assumption , that is what S(k) was assumed to be !

and the second term comes from subing n=k+1 into the general term on the LHS of the summation

Answer 11

now if you get S(k+1) over a single denominator and simplify , you will get S(k+1) = [ (k+1)/(k+2)] , which is what we set out to prove from step 3.

Hence it is proved by induction

the end

Answer 12

oh, my! thank you! I would never have gotten this. Thank you for explaining it step by step. SOOOOOO helpful!