Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 4.0 AMC 10/12 Find the answer China

Express the polynomial in $x$ $f(x) = 1 - x + x^2 - x^3 + \cdots - x^{19} + x^{20}$ into a polynomial in $y$ $g(y) = a_0 + a_1 y + a_2 y^2 + \cdots + a_{19} y^{19} + a_{20} y^{20}$ , where $y = x - 4$ . Then $\text{a 0 + a 1 + + a 20 =}$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

The terms in the expression $f(x)$ form a geometric series with first term $1$ and common ratio $-x$ . By the summation formula of geometric series,
$f(x) = \frac{(-x)^{21} - 1}{-x - 1} = \frac{x^{21} + 1}{x + 1}.$
Set $x = y+4$ , $g(y) = \frac{(y+4)^{21} + 1}{y+5}$ . Let $y = 1$ , we get
$a_0 + a_1 + \cdots + a_{20} = g(1) = \frac{5^{21} + 1}{6}.$

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