Menu Close

Let-f-and-g-be-functions-such-that-for-all-real-number-x-and-y-g-f-x-y-f-x-x-y-g-y-Find-the-value-of-g-0-g-1-g-2-g-3-g-2016-




Question Number 4702 by 314159 last updated on 22/Feb/16
Let f and g be functions such that for all  real number x and y,g(f(x+y))=f(x)+(x+y)g(y).  Find the value of g(0)+g(1)+g(2)+g(3)+...+g(2016).
$$\mathrm{Let}\:\mathrm{f}\:\mathrm{and}\:\mathrm{g}\:\mathrm{be}\:\mathrm{functions}\:\mathrm{such}\:\mathrm{that}\:\mathrm{for}\:\mathrm{all} \\ $$$$\mathrm{real}\:\mathrm{number}\:\mathrm{x}\:\mathrm{and}\:\mathrm{y},\mathrm{g}\left(\mathrm{f}\left(\mathrm{x}+\mathrm{y}\right)\right)=\mathrm{f}\left(\mathrm{x}\right)+\left(\mathrm{x}+\mathrm{y}\right)\mathrm{g}\left(\mathrm{y}\right). \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{g}\left(\mathrm{0}\right)+\mathrm{g}\left(\mathrm{1}\right)+\mathrm{g}\left(\mathrm{2}\right)+\mathrm{g}\left(\mathrm{3}\right)+…+\mathrm{g}\left(\mathrm{2016}\right). \\ $$
Commented by prakash jain last updated on 22/Feb/16
Let us try for trivial solution  f(x)=g(x)=0  Trivial solution satisfies the given condition  for all x,y∈R  so Σ_(i=0) ^(2016) g(i)=0  If there is a unique solution for the sum in  question then it must be equal to 0.
$$\mathrm{Let}\:\mathrm{us}\:\mathrm{try}\:\mathrm{for}\:\mathrm{trivial}\:\mathrm{solution} \\ $$$${f}\left({x}\right)={g}\left({x}\right)=\mathrm{0} \\ $$$$\mathrm{Trivial}\:\mathrm{solution}\:\mathrm{satisfies}\:\mathrm{the}\:\mathrm{given}\:\mathrm{condition} \\ $$$$\mathrm{for}\:\mathrm{all}\:{x},{y}\in\mathbb{R} \\ $$$$\mathrm{so}\:\underset{{i}=\mathrm{0}} {\overset{\mathrm{2016}} {\sum}}{g}\left({i}\right)=\mathrm{0} \\ $$$$\mathrm{If}\:\mathrm{there}\:\mathrm{is}\:\mathrm{a}\:\mathrm{unique}\:\mathrm{solution}\:\mathrm{for}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{in} \\ $$$$\mathrm{question}\:\mathrm{then}\:\mathrm{it}\:\mathrm{must}\:\mathrm{be}\:\mathrm{equal}\:\mathrm{to}\:\mathrm{0}. \\ $$

Leave a Reply

Your email address will not be published. Required fields are marked *