Mathematics

If $$f(x)$$ is odd function and $$f(1)=a$$, and $$f(x+2)=f(x)+f(2)$$ then the value of $$f(3)$$ is 


ANSWER

$$3a$$


SOLUTION
Given that $$f(x)$$ is an odd function and $$f(1)=a$$.....(1) and 

$$f(x+2)=f(x)+f(2)$$.....(2).

Since $$f(x)$$ is an odd function then we've, $$f(-x)=-f(x)$$ for all $$x$$.
Then we've, $$f(-1)=-f(1)=-a$$.......(3).

Now from (2) we get,

$$f(-1+2)=f(-1)+f(2)$$
or, $$f(1)=f(-1)+f(2)$$
or, $$a=-a+f(2)$$ [ Using (1) and (3)]
or, $$f(2)=2a$$......(4).

Now again from (2) we've,

$$f(1+2)=f(1)+f(2)$$
or, $$f(3)=a+2a$$ [ Using (1) and (4)]
or, $$f(3)=3a$$.
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Single Correct Medium Published on 17th 09, 2020
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