A Quintic Polynomial!

Algebra Level 5

Let $f(x)$ be a quintic polynomial such that if $n$ is a positive integer consisting of the only digit 7 repeated $k$ times, then $f(n)$ consists of the only digit 7 repeated $5k+3$ times.

For example : $f(77) = 7777777777777$ .

Then find the value of $f(1)$ to three correct places of decimals.

The answer is 48524.188.

1 solution

Patrick Corn
Aug 17, 2015

I get $f(x) = 1000 \cdot \frac79[(9x/7+1)^5-1] + 777$ . This is because the two sides agree on all $x$ whose only digits are $7$ , and they're both quintic polynomials, so they have to be the same since they agree in more than 5 places. (Try plugging in $x = 77\cdots 7$ to the right side; it's pretty clear what happens.)

So $f(1) = 116506577/2401 \equiv \fbox{48524.188}$ .

A Quintic Polynomial!

The answer is 48524.188.

1 solution

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