Geometric Progression

Question

Given that 1 + e^-3x + e^-6x + ..., find the value of x if S_∞ = 4. Give answer correct to four significant figures.

Answer 1

formula for the sum of an infinite geometric series = a1/(1-r)

a1 = 1st term

r = the common ratio = a term divided by its previous term

e^-3x/1 = e^-3x

e^-6x/e^-3x = e^(-6+3)x = e^-3x

r= e^-3x

sum = 4 = 1/(1-e^-3x)

1 - e^-3x = 1/4

e^-3x = 1-1/4 = 3/4

take natural logs of both sides to get

-3x = ln(3/4)

divide both sides by -3

then use a calculator with an exponent or log function

x =-ln(3/4)/3 = 0.095894024

round up for the 4th decimal place, 8 to 9

x = about 0.0959