QJEEM23Mac4c2: JEE Main 2023 Mathematics - Class 11 JEEM23Mac4c2 Solution: Consider a function , satisfying with . Then is equal to

$f(1) + 2f(2) + 3f(3)\, + \,...\, + \,nf(n) = n(n + 1) + (n)$ ..... (i)

$n \to n + 1$

$f(1) + 2f(2)\, + \,...\, + \,(n + 1)f(n + 1) = (n + 1)(n + 2)f(n + 1)$ ...... (ii)

(i) and (ii) gives

$3f(3) - 2f(2) = 0$

$4f(4) - 3f(3) = 0$

$\vdots$

$(n + 1)f(n + 1) - nf(n) = 0$

$\Rightarrow f(n + 1) = {{2f(2)} \over {n + 1}}$

$f(n) = {1 \over {2n}}$

${1 \over {f(2022)}} + {1 \over {f(2028)}} = 8100$