Use the formula from the video to simplify 1 + 1/4 + 1/16 + ... + 1/4^6

Question

include at least three decimal places in your answer. specific topic- Geometric Sequence and Future Value of Annuities

Answer 1

common ratio = 1/4 = r, general formula for the sum is

derived by multiplying S by r and subtracting

S = a + ar + ... + ar^6

rS=ar + .....ar^7

rS-S = ar^7 - a

S = a(r^7-1)/(r-1)

for a=1 and r=1/4

S = ((1/4)^7 - 1)/(1/4 - 1) = (4/3)(1-1/4^7) = (4/3)(16383/16384) = 1.333252

1.333 for 3 decimal places. It's virtually 4/3, 4/3 = 1.333 rounded to 3 decimal places

1+1/4 = 1.25

1.25+ 1/16 = 1.3125

1.3125 + 1/64 = 1.328125

....

1.333 = 4/3 for n=infinity S=1+r/(1-r) = 1+(1/4)/(3/4) = 1+(1/4)(4/3) =1+ 1/3 = 4/3

the more terms you add the closer you get to 4/3