Question

Find the sum of the geometric sequence. 1, 1/4, 1/16, 1/64, 1/256

A) 341

B) 1/192

C) 1/768

D) 341/256

Answer 1

`1,\ (1)/(4);\ (1)/(16);\ (1)/(64);\ (1)/(256)\ \text{is geometric sequence}\\\\\text{the ratio is:}\ r=((1)/(4))/(1)=(1)/(4)\\\\\text{The sum of the numbers in a geometric sequence is:}\\\\S_n=(a_1(1-r^n))/(1-r)\\\\a_1=1;\ r=(1)/(4)\\\\\text{substitute}\\\\S_5=(1\cdot\left[1-\left((1)/(4)\right)^5\right])/(1-(1)/(4))=(1-(1)/(1024))/((3)/(4))=(1023)/(1024)\cdot(4)/(3)=(341)/(256)\to D)`

Other method:

`1=(256)/(256)\\\\(1)/(4)=(1\cdot64)/(4\cdot64)=(64)/(256)\\\\(1)/(16)=(1\cdot16)/(16\cdot16)=(16)/(256)\\\\(1)/(64)+(1\cdot4)/(64\cdot4)=(4)/(256)\\\\1+(1)/(4)+(1)/(16)+(1)/(64)+(1)/(256)=(256)/(256)+(64)/(256)+(16)/(256)+(4)/(256)+(1)/(256)=(341)/(256)\to D)`