Answer:
21 terms
Step-by-step explanation:
The sum to n terms of a geometric progression is
[tex]S_{n}[/tex] = [tex]\frac{a(r^n-1)}{r-1}[/tex]
where a is the first term and r the common ratio
Here a = 0.5 and r = 1 ÷ 0.5 = 2
Equate the sum to 1000000 and solve for n , that is
[tex]\frac{0.5(2^n-1)}{2-1}[/tex] = 1000000
0.5([tex]2^{n}[/tex] - 1) = 1000000 ( divide both sides by 0.5 )
[tex]2^{n}[/tex] - 1 = 2000000 ( add 1 to both sides )
[tex]2^{n}[/tex] = 2000001 ( take the ln of both sides )
ln[tex]2^{n}[/tex] = ln(2000001)
n ln2 = ln(2000001) ( divide both sides by ln 2 )
n = [tex]\frac{ln2000001}{ln2}[/tex] = 20.9315....
Thus the number of terms to exceed 1000000 is n = 21
