If-f-x-f-x-1-x-2-2x-and-f-0-17-find-f-17-

Question Number 164396 by ajfour last updated on 16/Jan/22

I f f (x) = f (x - 1) + x^{2} + 2 x

a n d f (0) = 17, f i n d f (17) .

Commented by ajfour last updated on 16/Jan/22

T h a n k y o u s i r s .

Answered by mahdipoor last updated on 16/Jan/22

f (1) = f (0) + 1^{2} + 2 \times 1

f (2) = f (1) + 2^{2} + 2 \times 2

f (3) = f (2) + 3^{2} + 2 \times 3

\dots

f (17) = f (16) + 17^{2} + 2 \times 17

\Rightarrow f (1) + \dots + f (17) =

f (0) + \dots + f (16) + (1^{2} + \dots + 17^{2}) + 2 (1 + \dots + 17)

\Rightarrow f (17) = f (0) + \frac{17 \times 18 \times 35}{6} + 2 \frac{17 \times 18}{2}

\Rightarrow f (17) = 2108

Answered by mr W last updated on 16/Jan/22

f (k) - f (k - 1) = {(k + 1)}^{2} - 1

\underset{k = 1}{\sum^{n}} [f (k) - f (k - 1)] = \underset{k = 1}{\sum^{n}} {(k + 1)}^{2} - n

f (n) - f (0) = \underset{k = 2}{\sum^{n + 1}} k^{2} - n

f (n) = f (0) + {(n + 1)}^{2} - 1 + \frac{n (n + 1) (2 n + 1)}{6} - n

f (n) = f (0) + \frac{n (n + 1) (2 n + 7)}{6}

f (x) = f (0) + \frac{x (x + 1) (2 x + 7)}{6}

\Rightarrow f (x) = 17 + \frac{x (x + 1) (2 x + 7)}{6}

\Rightarrow f (17) = 17 + \frac{17 \times 18 \times 41}{6} = 2108

Commented by Rasheed.Sindhi last updated on 16/Jan/22

\lor \cap i \subset\in Sir!

\lor \cap \boldsymbol i \subset\in Sir!

Commented by Tawa11 last updated on 16/Jan/22

Great sirs

Great sirs

Facebook Tweet Pin