Question

Fie functia f:R R , f(x)=2×-3. Sa se calculeze f(1)+f(2)+...+f(100)

Answer 1

$f:\mathbb{R}\rightarrow\mathbb{R} \: ,f(x) = 2x - 3$

$f(1) + f(2) + ... + f(100)$

$f(1) = 2 \times 1 - 3 = - 1$

$f(2) = 2 \times 2 - 3 = 1$

$f(3) = 2 \times 3 - 3 = 3$

$f(4) = 2 \times 4 - 3 = 5$

$f(100) = 2 \times 100 - 3 = 197$

$f(1) + f(2) + f(3) + f(4) + ... + f(100)$

$= - 1 + 1 + 3 + 5 + ... + 197$

$= 3 + 5 + ... + 197$

$3 = 2 \times 1 + 1$

$5 = 2 \times 2 + 1$

$197 = 98 \times 2 + 1$

$3 + 5 + ... + 197 =$

$= 2 \times 1 + 1 + 2 \times 2 + 1 + ... + 2 \times 98 + 1$

$= 2(1 + 2 + ... + 98) + 98$

$= 2 \times \frac{98(98 + 1)}{2} + 98$

$= 98(98 + 1) + 98$

$= 98 \times 99 + 98$

$= 98(99 + 1)$

$= 98 \times 100$

$= 9800$