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Let a,b,c in R. " If " f(x) =ax^(2)+bx+c...

Let `a,b,c in R. " If " f(x) =ax^(2)+bx+c` be such that `a+b+c=3 and f(x+y)=f(x)+f(y)+xy, AA x,y in R, " then " sum_(n=1)^(10)f(n)` is equal to

A

330

B

165

C

190

D

255

Text Solution

Verified by Experts

The correct Answer is:
A
We have, `f(x)=ax^(2)+bx+c`
Now, `f(x+y)=f(x) +f(y)+xy`
Put `y=0 rArr f(x) = f(x)+f(0)+0`
`rArr f(0)=0`
`rArr c=0`
Again, put `y= -x`
` therefore f(0)=f(x)+f(-x)-x^(2)`
`rArr 0=ax^(2)+bx+ax^(2)-bx-x^(2)`
`rArr 2ax^(2)-x^(2)=0`
`rArr a=(1)/(2)`
Also, `a+b+c=3`
`rArr (1)/(2) +b+0=3 rArr b=(5)/(2)`
` therefore f(x)=(x^(2)+5x)/(2)`
Now, `f(n)=(n^(2)+5n)/(2)=(1)/(2)n^(2)+(5)/(2)n`
`therefore sum_(n=1)^(10)f(n)=(1)/(2)sum_(n=1)^(10)n^(2)+(5)/(2)sum_(n=1)^(10)n`
`=(1)/(2)*(10xx11xx21)/(6)+(5)/(2)xx(10xx11)/(2)`
`=(385)/(2)+(275)/(2)=(660)/(2)=330`
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