Question

Answer and process,pls,very urgent

Answer 1

f : N ———>R be a function satisfying following conditions :
f( 1) = 1

f(1) + 2f(2) + 3f(3) +....nf(n ) = n( n +1)f( n)

for n = 2

f(1) +2 f(2) = 2( 2 + 1) f(2)

f(1) + 2f(2) = 6f(2)

f(2) = f(1)/4 = 1/4

for n = 3

f(1) + 2f(2) + 3f(3) = 3(3+1) f(3)

f(1) + 2× f(1)/4 + 3f(3) = 12 f(3)

1 + 1/2 = 9f(3)

3/2 = 9f(3)

f(3) = 1/6

for n = 4

f(1) + 2f(2) + 3f(3) + 4f(4) = 4×5 f(5)

1 + 2× 1/4 + 3× 1/6 + 4 f(4) = 20f(5)

1 + 1/2 + 1/2 = 16f(4)

2 = 16f(4)

f(4) = 1/8

now, we conclude that ,

1 × f(1) = 1
2 × f(2) = 1/2
3 × f(3) = 1/2
4 × f(4) = 1/2

hence,

f(1) +2f(2) + 3f(3) +.....nf(n) = 1 + 1/2 + 1/2 + 1/2 +....1/2 = n( n +1) f( n)

1 + n { 1/2 } = n( n +1)f( n)

1 + n/2 = n( n +1)f( n)

f( n) = ( n +2)/2n( n +1)

nf( n) = (n +2)/2( n +1)

put n = 49

49f( 49) = (49+2)/2×(49+1) = 51/100