Let f(x) be a function such that f(x + y) = f(x) + f(y), for all real x and y. If it is given that f(11) = 55, what is f(1) + f(2) + f(3) + … + f(10)?
Explanation:
Given, f(x + y) = f(x) + f(y)
Substitute y = 0
⇒ f(x + 0) = f(x) + f(0)
⇒ f(0) = 0
Now, substitute x = y = 1, we get
f(1 + 1) = f(1) + f(1)
⇒ f(2) = 2f(1)
Now, substitute x = 2 and y = 1, we get
f(2 + 1) = f(2) + f(1) = 2f(1) + f(1) = 3f(1)
∴ f(3) = 3f(1)
Now, substitute x = 3 and y = 1, we get
f(3 + 1) = f(3) + f(1) = 3f(1) + f(1) = 4f(1)
∴ f(4) = 4f(1)
By observing the pattern, we can say that
f(x) = x × f(1)
∴ f(11) = 11 × f(1) = 55
⇒ f(1) = 5
Now, we have to calculate the value of f(1) + f(2) + f(3) + … + f(10)
= f(1) + 2f(1) + 3f(1) + … + 11f(1)
= f(1) × [1 + 2 + 3 + … + 11]
= 5 × 55
= 275
Hence, 275.
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