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If f(x) = (484)^(x-1) / ((484)^x + 22), find the value of f(1/45) + f(2/45) + f(3/45) +... + f(44/45).

1. 44
2. 22
3. 1/11
4. 1/22

Correct answer: 1/22

Solution

Let a = 484^x. f(x) = (484)^(x-1)/((484)^x + 22) = a/(484*(a+22)) = a/(484a + 484*22). Actually: f(x) = 484^(x-1)/(484^x + 22) = (484^x / 484)/(484^x + 22) = a/(484(a+22)/484)... Let me redo: f(x) = 484^(x-1)/(484^x + 22). Let t = 484^x. Then f(x) = t/484/(t+22) = t/(484t + 484*22)? No: f(x) = (t/484)/(t+22) = t/(484(t+22)). f(1-x) = 484^(-x)/(484^(1-x)+22) = (1/t)/((484/t)+22) = (1/t)/((484+22t)/t) = 1/(484+22t). f(x)+f(1-x) = t/(484(t+22)) + 1/(484+22t) = t/(484(t+22)) + 1/(22(22+t)) [since 484=22², 484+22t=22(22+t)]. = t/(484(t+22)) + 1/(22(t+22)) = [t/484 + 1/22]/(t+22) = [t + 22]/(484(t+22)) [since 1/22 = 22/484] = 1/484... Recheck: t/484 + 1/22 = t/484 + 22/484 = (t+22)/484. So f(x)+f(1-x) = (t+22)/(484(t+22)) = 1/484... Hmm that gives 1/484 not 1/22. Let me recalculate more carefully.

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