Question

Hello! I don't know why I am getting the wrong answer, please help!
10 .The first three terms of a geometric sequence are 100, 90 and 81.
(iv) After how many terms is the sum of the sequence greater than 99% of the sum to infinity.
I get -22. the answer according to the text book is 44. In a previous question it is shown that the r=9/10

Answer 1

oh I forgot to mention the sum to infinity of the terms of the sequence is 1000

Answer 2

Did you figure out the general sequence yet using those terms given

Answer 3

oh ok, had to look what that meant, sum to infinity is the limit of that sequence

Answer 4

have you seen this before:

\[S _{n} = \frac{ a _{1}(1-r^n) }{ 1-r }\]

sum of n terms of geometric series

Answer 5

r = ratio of n+1 term to the nth term ...

r = 90/100 = 81/90 = 0.9

Answer 6

since the terms are always getting smaller and smaller by that ratio,, eventually the thing will settle or 'converge' to a certain value, the farther out you go in the value of n, the closer the total will be to the limiting value of the series

Answer 7

sorry back,

Answer 8

yeah 44 looks right

Answer 9

since they just gave you the sum to infinity is 1000, 99% of that is 990

you want that series to total 990 or more and find out how many terms it takes, n

Answer 10

\[990=\frac{ 100(1-0.9^n )}{ 1-0.9 }\]

solve

Answer 11

you see the value for n, round up to the next whole number, that is the number of terms to get to 990, or , 99% of 1000

Answer 12

i got 43.something, so 44 terms to get there

Answer 13

Oh I think I see. Thank you very much!

Answer 14

yes yes, if they dont give you a value for the limit or infinite sum, then you can find it by the first value in the sequence divided by (1 - r)

100/(1-0.9) = 1000

Answer 15

if r is larger than 1, then the thing will not have an infinite sum or converging value as n goes to larger

Answer 16

in that case it just blows up since each term is larger than the last

Answer 17

Ok. Good to know. :) Thanks very much!