Question

If `f(1)=1 and f(1)+2f(2)=3f(3)+......nf(n)= n(n+1)f(n)` Then find the value of `49f(49)`

Answer 1

`f(1) = 1`
`f(1)+2f(2)+3f(3)+...+nf(n) = n(n+1)f(n)`
`:. f(1)+2f(2) = 2*3*f(2)`
`=>f(1) = 4f(2)`
`=>1 = 4f(2)`
`=>2f(2) = 1/2`
Now, `f(1)+2f(2)+3f(3) = 3*4*f(3)`
`=>1+1/2 = 9f(3)`
`=>3/2 = 9f(3)`
`=>3f(3) = 1/2`
Now, `f(1)+2f(2)+3f(3)+4f(4) = 4*5*f(3)`
`=>1+1/2 +1/2+ 4f(4) = 20f(4)`
`=>2 = 16f(4)`
`=>4f(4) = 1/2`
Now, `f(1)+2f(2)+3f(3)+4f(4)+5f(5) = 5*6*f(3)`
`=>1+1/2 +1/2+ +1/2 + 5f(5) = 30f(4)`
`=>5/2 = 25f(5)`
`=>5f(5) = 1/2`
So, we can see that, `nf(n) = 1/2.`
`:. 49f(49) = 1/2`