Let f be a polynomial function such that , x 0 and f(6) = 37. Then is equal to ________. [2026]

Question

Let f be a polynomial function such that $\log_{2} (f (x)) = (\log_{2} (2 + \frac{2}{3} + \frac{2}{9} + . . . \infty)) \cdot \log_{3} (1 + \frac{f (x)}{f (1 / x)})$ , x > 0 and f(6) = 37. Then $\sum_{n = 1}^{10} f (n)$ is equal to ________. [2026]

Smart Achievers · Accepted Answer

(395)

$\log_{2} (f (x)) = (\log_{2} (2 + \frac{2}{3} + \frac{2}{9} + . . . \infty)) \cdot \log_{3} (1 + \frac{f (x)}{f (1 / x)})$

$S = \frac{a}{1 - r} \Rightarrow S = \frac{2}{1 - 1 / 3} = 3$

$\log_{2} (f (x) = \log_{2} (3) \cdot \log_{3} (1 + \frac{f (x)}{f (1 / x)})$

Using base change formula

$\log_{2} (3) \cdot \log_{3} (A) = \log_{2} (A)$

$\Rightarrow \log_{2} (f (x) = \log_{2} (1 + \frac{f (x)}{f (1 / x)})$

$\Rightarrow f (x) = 1 + \frac{f (x)}{f (1 / x)}$

$\Rightarrow f (x) - 1 = \frac{f (x)}{f (1 / x)}$

$\Rightarrow f (1 / x) = \frac{f (x)}{f (x) - 1} 3$

$\Rightarrow f (x) \cdot f (1 / x) = f (x) + f (1 / x)$

$\Rightarrow f (x) = 1 \pm x^{n} \Rightarrow f (6) = 37$

If $f (x) = 1 - x^{n}$ , then $1 - 6^{n} = 37 \Rightarrow - 6^{n} = 36$ (no real solution)

If $f (x) = 1 + x^{n}$ , then $1 + 6^{n} = 37 \Rightarrow 6^{n} = 36 \Rightarrow n = 2$

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

$\sum_{n = 1}^{10} f (n) = \sum_{n = 1}^{10} (n^{2} + 1) = \sum_{n = 1}^{10} n^{2} + \sum_{n = 1}^{10} 1$

$= \underset{for n = 10}{\underset{⏟}{\frac{n (n + 1) (2 n + 1)}{6} + 10}}$

$= \frac{10 \times 11 \times 21}{6} + 10 = 385 + 10 = 395$

Solution

Q.
Let f be a polynomial function such that $\log_{2} (f (x)) = (\log_{2} (2 + \frac{2}{3} + \frac{2}{9} + . . . \infty)) \cdot \log_{3} (1 + \frac{f (x)}{f (1 / x)})$ , x > 0 and f(6) = 37. Then $\sum_{n = 1}^{10} f (n)$ is equal to ________. [2026]

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Solution

Q. Let f be a polynomial function such that log2(f(x))=(log2(2+23+29+...∞))·log3(1+f(x)f(1/x)), x > 0 and f(6) = 37. Then ∑n=110f(n) is equal to ________. [2026]

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Q.
Let f be a polynomial function such that $\log_{2} (f (x)) = (\log_{2} (2 + \frac{2}{3} + \frac{2}{9} + . . . \infty)) \cdot \log_{3} (1 + \frac{f (x)}{f (1 / x)})$ , x > 0 and f(6) = 37. Then $\sum_{n = 1}^{10} f (n)$ is equal to ________. [2026]