Question

What is the sum, of 'n' terms in the series : log m + log \(\big(\frac{m^2}{n}\big)\) + log \(\big(\frac{m^3}{n^2}\big)\) + log \(\big(\frac{m^4}{n^3}\big)\) + .......

(a) log \(\bigg[\frac{n^{(n-1)}}{m^{(n+1)}}\bigg]^{\frac{n}{2}}\)

(b) log \(\bigg[\frac{m^m}{n^n}\bigg]^{\frac{n}{2}}\) 

(c) log \(\bigg[\frac{m^{(1-n)}}{n^{(1-m)}}\bigg]^{\frac{n}{2}}\) 

(d) log \(\bigg[\frac{m^{(1+n)}}{n^{(n-1)}}\bigg]^{\frac{n}{2}}\) 

Answer 1

(d) log \(\bigg[\frac{m^{(1+n)}}{n^{(n-1)}}\bigg]^{\frac{n}{2}}\) 

S = log m + log \(\frac{m^2}{n}\) + log \(\frac{m^3}{n^2}\) + ...........n terms

= log \(\bigg[\)m.\(\frac{m^2}{n}\).\(\frac{m^3}{n^2}\)..........\(\frac{m^3}{n^{n-1}}\)\(\bigg]\) = log \(\bigg[\)\(\frac{m^{(1+2+3+4+......+n)}}{n^{(1+2+3+.....(n-1))}}\)\(\bigg]\)

= log \(\bigg[\)\(\frac{m^{\frac{n(n+1)}{2}}}{n^{\frac{n(n-1)}{2}}}\)\(\bigg]\) =  log \(\bigg[\frac{m^{(1+n)}}{n^{(n-1)}}\bigg]^{\frac{n}{2}}\)