If the value of an article is assumed to increase annually by 5% of its value at the beginning of the year,after how many years will its value double?

Question

a)If the value of an article is assumed to increase annually by 5% of its value at the beginning of the year,after how many years will its value double?

b).If the common ratio of a GP with a sum to infinity is x² -x -1,within what limits must x lie?

Answer 1

1.05ⁿ=2

nlog1.05=log2

n=log2/log1.05=14.20669908

I did the second part separately before.  Here it is
Assuming that x²-x-1≠1 the finite sum G=Σ_k=0^k=N-1(x²-x-1)k=(1-(x²-x-1)^N)/(1-(x²-x-1)) .
The only way that this sum converges as N→∞ is for (x²-x-1)^N→0 or that |x²-x-1|<1
To force this means that x²-x-1<1 and x²-x-1>-1
x²-x-1<1 means that x²-x-2<0.  That is between the roots of this parabola. x²-x-2=(x+1)(x-2)=0 between
x=-1 or x=2.  So on the interval -1<x<2.;  For the second condition x²-x-1>-1, which is x²-x>0, that is outside of the interval between the the roots of this parabola.  x²-x=x(x-1)=0 when x=0 or x=1. So on the intervals
(-∞,0) or (1,∞)
The intersection of these intervals is (-1<x<0)∪(1<x<2).  For x in this set the infinite sum converges.