f-x-x-20-a-1-x-19-a-2-x-18-a-20-If-f-1-f-2-f-3-f-20-What-is-the-value-of-a-1-

Question Number 33571 by Joel578 last updated on 19/Apr/18

f (x) = x^{20} + a_{1} x^{19} + a_{2} x^{18} + \dots + a_{20}

If f (1) = f (2) = f (3) = \dots = f (20)

What is the value of a_{1} ?

Answered by MJS last updated on 19/Apr/18

let me downsize this

f (x) = x^{2} + a_{1} x + a_{2}

f (1) = 1 + a_{1} + a_{2}

f (2) = 4 + 2 a_{1} + a_{2}

f (2) - f (1) = 0

3 + a_{1} = 0 \Rightarrow a_{1} = - 3

f (x) = x^{3} + a_{1} x^{2} + a_{2} x + a_{3}

f (1) = 1 + a_{1} + a_{2} + a_{3}

f (2) = 8 + 4 a_{1} + 2 a_{2} + a_{3}

f (3) = 27 + 9 a_{1} + 3 a_{2} + a_{3}

f (3) - f (2) = 0

19 + 5 a_{1} + a_{2} = 0 (i)

f (2) - f (1) = 0

7 + 3 a_{1} + a_{2} = 0 (i i)

(i) - (i i)

12 + 2 a_{1} = 0

a_{1} = - 6

f (x) = x^{4} + \dots \Rightarrow

\Rightarrow a_{1} = - 10

f (x) = x^{5} + \dots \Rightarrow

\Rightarrow a_{1} = - 15

f (x) = x^{n} + \dots \Rightarrow

\Rightarrow a_{1} = - \frac{n}{2} (n + 1)

\Rightarrow f (x) = x^{20} + \dots \Rightarrow a_{1} = - 210

Commented by Joel578 last updated on 20/Apr/18

t h a n k y o u v e r y m u c h

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