Question

log(x/x+1)+log(x+1/x+2).. to n terms=​

Answer 1

Question: log(x/x+1) + log(x+1/x+2) + log(x+2/x+3) ... to n terms =

Answer:

$\sf{log\large{\frac{x}{x+n}}}$

Step-by-step explanation:

$\sf{Using \: the \: property \: of \: logarithms}$

$\bold {\sf{ log \frac{a}{b} = loga \: - \: logb} }$

$\sf{Therefore, }$

$\sf{ \small{log \frac{x}{x + 1} + log \frac{x + 1}{x + 2} + log \frac{x + 2}{x + 3} + ... \: n \: terms \: } }$

$\sf{ \small{log \frac{x}{x + 1} + log \frac{x + 1}{x + 2} + log \frac{x + 2}{x + 3} + ... + log \frac{x + (n - 2)}{x + (n - 1)} + log \frac{x + (n - 1)}{x + n} } }$

$\sf{log x - log(x + 1) + log(x + 1) - log(x + 2) - log(x + 3) + ...+log(x + (n - 2)) - log(x + (n - 1)) + log(x + (n - 1)) - log(x + n) \: \: }$

$\sf{logx - log(x + n)}$

$\sf{log \large{\frac{x}{x + n}} \: }$