A function f(x) is defined as f(x) = f(x - 2) - x(x + 2) for all the integer values of x and f(1) + f(4) = 0. What is the value of f(1) + f(2) + f(3) + f(4) + f(5) + f(6)?
Answers
Answer:
Step-by-step explanation:
f(1)+f(2)+f(3)+f(4)+f(5)+f(6)
Given, f(1)+f(4)=0
f(x)=f(x-2)-x(x-2)
f(2)+f(3)+f(5)+f(6)
f(2)=0
f(3)=f(1)-3
f(5)=f(3)-15
f(6)=f(4)-24
then, f(2)+f(3)+f(4)+f(6)=f(1)-3+f(3)-15+f(4)-24
=f(3)-42
=(f(1)-3)-42
=f(1)-45
=f(-1)-3-45
=48
Answer:
Let S = f(1) + f(2) + f(3) + f(4) + f(5) + f(6)
As f(1) + f(4) = 0, therefore S = f(2) + f(3) + f(5) + f(6) ------ (1)
f(2) = f(0) - 8
f(3) = f(1) - 15
f(4) = f(2) - 24 = f(0) - 32
f(5) = f(3) - 35 = f(1) - 50
f(6) = f(4) - 48 = f(0) - 80
Put the above values in equation (1), we get
S = f(0) - 8 + f(1) - 15 + f(1) - 50 + f(0) - 80
S = 2(f(0) + f(1)) - 153 ------ (2)
As we already know f(1) + f(4) = 0 ⇒f(1) + f(0) - 32 = 0 ⇒f(1) + f(0) = 32
Putting this value in equation 2, we get S = 2(32) - 153 = -89
So, Ans is -89