Proof?(functions)

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May 20th 2009, 09:01 PM
fardeen_gen

Proof?(functions)

Consider a real valued function $f(x)$ satisfying, $2f(xy) = (f(x))^y + (f(y))^x$ for all real $x\ \mbox{\&}\ y$ and $f(1) = a$ where $a\neq 1$ . Prove that $(a - 1)\sum_{i = 1}^{n}f(i) = a^{n + 1} - a$ .
May 21st 2009, 04:19 AM
HallsofIvy

Quote:

Originally Posted by fardeen_gen

Consider a real valued function $f(x)$ satisfying, $2f(xy) = (f(x))^y + (f(y))^x$ for all real $x\ \mbox{\&}\ y$ and $f(1) = a$ where $a\neq 1$ . Prove that $(a - 1)\sum_{i = 1}^{n}f(i) = a^{n + 1} - a$ .

Taking y= 1, $2f(x)= f(x)+ f(1)^x= f(x)+ a^x$ . That is, $f(x)= a^x$ . That should be all you need.

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