Question

I know I'm just overthinking this and it's literally probably an x=y formula but humor me.

Midas wants to buy all the land near his house. There are 65 lots surrounding his home, and each lot costs $7200. But for every lot purchased, the price doubles. How much money does Midas need to buy all 65 lots?

Answer 1

Use sum of GP series here

Answer 2

a(R^n-1) / R-1

here a= 7200
n=65
and r=2

Answer 3

it says for every lot purchased, the price doubles
you can try without the fancy geometric series by using simple arithmetic first
Price per lot = 7200
number of lots 65
normally it would be 65* 7200 but the condition is that the price increases for every additional lot purchased
so if we start with the first lot \(L_1 = 7200\) then second lot \(L_2 = 2 \times L_1\) then third \(L_3 = 2 \times L_2 \)
we can quickly see that we are doubling the previous price

Answer 4

so maybe it would be safe to say that our addition of lot sequence would be \(\large L_1 + L_2 + L_3 ... L_{65}\)
we will worry about inserting each of \(L_n \) prices later and focus on how we can write L into condensed form. Have you done any summations in the past?

Answer 5

humour me

Answer 6

Let's pretend for a second there are only 3 lots. Then the prices for buying will be:

\[7200 + 2*7200+ 2*2*7200\]
Which simplifies to
\[7200(1+2+4)\]

So this is where the geometric series comes in, instead of only having 3, we have 65 which will be a geometric series in 2 which is exactly what @arindameducationusc is referring to here.

Answer 7

Nice @Kainui & @nincompoop...
Good job!