Geometric series for this problem..?

Question

I am trying to work this out with geometric series:

   A certain Australian politician claimed that children born today may live to age 150. Is
this claim reasonable?
In the year 2000 average life expectancy in Australia was 78 years. For the purpose of this question, let
us assume that a person aged x ≤ 78 in the year 2000 would live a further 78 − x years (on average).
Because of healthcare improvements life expectancy in developed countries has been increasing by 2.5
years per decade. Assume this trend continues indefinitely into the future.
(a) How long can someone born in Australia in the year 2000 expect to live, on average?
(b) What is the smallest value of y such that someone born in the year 2000+y would live to be 150
on average?

I have been trying to work out a geometric series equation for this, but I cant seem to get it right.

I was just wondering if someone wouldnt mind explaing how i would find the formula for this?

Thanks C :)

Answer 1

i am going to make following assumption:

life expectancy of someone born on the given year is how long the person can be expected to live on average, when the person is born on the given year. Meaning future life expectancy increase do not impact person already born. (i think that is what the problem assumption is)

given above assumption:

(1) by life expectancy definition, people born in 2000 can expect to live to 78

(2) given 2.5 year increase in life expectancy per 10 year, then it would take 29 decades to achieve 72.5 year increase in life expectancy. (72.5+78) = 150.5 year.

it would take 290 year from 2000 to have 150 for life expectancy

edit note:

thank you for additional info.

given that people already also born also receive benefit to increase to life expectancy, then following is solution:

life expectancy remaining for someone born in 2000:

2000, remaining life = 78 2010, remaining life = 78-10+2.5 = 78-7.5 2020, remaining life = 78-7.5-7.5 = 78-7.5*2 ... person is expected die when remaining life = 0. the effective life decreasing rate is 7.5year/decade, meaning it would take 10 decades to use up 75 remaining life. thus, someone born in 2000 is expected to die after 10 decade + 3 year = 103year

for someone to die after 150 year, let x = life expectancy when born, then

x/7.5 = 150/10, which mean x = 112.5.

thus, when someone is born with 112.5 year of life expectancy, he is expected to live to 150