1)prove that logx+log(xy)+log(xy^2)+...is an arithmetical progression and show,without using tables that when x=160

Question

2)sum the series 2+6+18+54... To thirty term,by using logarithm,verify that this sumis breather than 2x10^14

Answer 1

well,
log(xy) = logx + logy
so, you have an AP with
a = logx
d = logy

S30 = 2(3^30-1)/(3-1) ≈ 3^30 (10^.477)^30 ≈ 10^14.313

log2 = .3010, so

10^.313 * 10^14 > 2*10^14