Question

The first term of an infinite G.P is 1 and any term is equal to the sum of all succeeding terms. Find the series

Answer 1

\[\large S_{\infty} = \dfrac{1}{1-r}\]

Answer 2

you're given :

\[\large 1 = S_{\infty } - 1\]

Answer 3

you can solve \(r\), right ?

Answer 4

hey sorry i was afk

Answer 5

My textbook has a sollution i don't get it

Answer 6

\[\large a_n = S_{\infty} - S_{n}\]

Answer 7

Let the GP be 1,r , r^2 , r^3 ........
give condition:-
\[r=\frac{ r ^{2} }{ 1-r }\]
Thus r = 1/2

Answer 8

i don't get how they got this
\[r=\frac{ r ^{2} }{ 1-r }\]

Answer 9

\[\frac{ a }{ 1-r }\] i know this

Answer 10

they have just taken part of the given constraint :
Notice that \(r\) is the second term of given GP

Answer 11

1, `r` , r^2 , r^3 ...

Answer 12

what does the constraint tell us ?

Answer 13

this second term equals sum of all the next terms, right ?

Answer 14

yes

Answer 15

r = r^2 + r^3 + ...

Answer 16

yes got it .

Answer 17

good :)