Question

If f : R `rightarrow` R satisfies f(x + y) = f(x) + f(y), for all x, y ∈ R and f(1) = 7, then `sum_(r = 1)^nf(r)` is ______.

विकल्प

• `(7n(n + 1))/2`

• `(7n)/2`

• `(7(n + 1))/2`

• 7n + (n + 1)

Answer 1

If f: R `rightarrow` R satisfies f(x + y) = f(x) + f(y), for all x, y ∈ R and f(1) = 7, then `sum_(r = 1)^nf(r)` is `underlinebb((7n(n + 1))/2)`.

Explanation:

Let x = 0 = y 

Therefore, f(0) = 0 and x = 1, y = 0

Therefore, f(1 + 0) = f(1) + f(0) = 7  ...(Given)

x = 1, y = 1 

Therefore, f(1 + 1) = 2f(1) = 2(7)

Therefore, f(2) = 2(7)

x = 1, y = 2

Therefore, f(3) = f(1) + f(2)

= 7 + 2(7)

= 3(7)

and so on

∴ `sum_(r = 1)^nf(r)` = f(1) + f(2) + f(3) + .... + f(n)

= 7(1 + 2 + 3 + ... + n)

= `(7n(n + 1))/2`