Binomial Theorem

Question
CBSEENMA11015543

Let a, b, c ∈ R. If f(x) = ax2 + bx + c is such
that a + b + c = 3 and f(x + y) = f(x) + f(y) + xy, ∀x,y ∈ R,then sum from straight n space equals 1 to 10 of space straight f space left parenthesis straight n right parenthesis is equal to

  • 225

  • 330

  • 165

  • 190

Solution

B.

330

f(x) = ax2 + 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) = f(x) + x + 3
Now
f(2) = 7
f(3) = 12
Now
Sn = 3 + 7 + 12 + ..... tn ...(1)
Sn = 3 + 7 + ......tn – 1 + tn ...(2)
On subtracting (2) from (1)
tn = 3 + 4 + 5 + ....... upto n termsstraight t subscript straight n space equals space fraction numerator left parenthesis straight n squared space plus 5 straight n right parenthesis over denominator 2 end fraction
straight S subscript straight n space equals space straight capital sigma space straight t subscript straight n space
equals space space stack sum space fraction numerator left parenthesis straight n squared space plus space 5 straight n right parenthesis over denominator 2 end fraction with space below and space on top
straight S subscript straight n space equals space 1 half space open square brackets fraction numerator straight n space left parenthesis straight n plus 1 right parenthesis left parenthesis 2 straight n plus 1 right parenthesis over denominator 6 end fraction plus fraction numerator 5 straight n space left parenthesis straight n plus 1 right parenthesis over denominator 2 end fraction close square brackets
straight S subscript 10 space equals space 330

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