\hspace{-16}$If $\bf{f(x+1)=(-1)^{x+1}.x-2f(x)\forall x\in \mathbb{N}}$ and $\bf{f(1)=f(1986)}$\\\\ Then Sum of Digit of the no. $\bf{f(1)+f(2)+f(3)+.....+f(1985)}$
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2 Answers
1357f(x+1)+f(x)=(-1)(x+1).x-f(x)
Now we have
A=f(1)+f(2)+f(3)......+f(1985)
A=(1/2)(2f(1)+2f(2)+2f(3)......+2f(1985)
A=(1/2)(f(1)+[f(1)+f(2)]+[f(2)+f(3)]......)
A=(1/2){[f(1)+f(2)]+f(2)+f(3)+......[f(1985)+f(1986)]
proceed and get the answer :)
21let S= f(1)+f(2)+.....
now,
f(2)=1-2f(1)
f(3)=-2-2f(2)
f(4)=3-2f(3)
.
.
.
f(1986)=1985-2f(1985)
Hence add all equations
hence,
S=(1-2+3...+1985) -2S
now compute S and compute the answer
