Test Index

Functions Part 1

Section: Mathematics

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Question : 79 of 100

Marks: +1, -0

Let a,b,c ∊ R.If

f(x)=ax^2+bx+c

a+b+c=3

f(x+y)=f(x)+f(y)+xy,\forall x,y\in R

\sum\limits_{n=1}^{10}f(n)

225
330
165
190

Solution:

f(x)=ax^2+bx+c

f(1)=a+b+c=3

Now

f(x+y)=f(x)+f(y)+xy

put

y=1

f(x+1)=f(x)+f(1)+x

f(x+1)+x+3

Now

f(2)=7

f(3)=12

Now

S_n=3+7+12+.....t_n

...........(1)

S_n=3+7+.....t_{n-1}+t_n

.........(2)

On subtracting (2) from (1)

t_n=3+4+5+....

upto n terms

t_n=\frac{n^2+5n}{2}

S_n\sum t_n = \sum \frac{n^2+5n}{2}

S_n=\frac{1}{2}\left[ \frac{n(n+1)(2n+1)}{6} + \frac{5n(n+1)}{2} \right]

S_{10}=330

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