1. ## geometric sequences

hi everyone!
I need help with these two problems. thanks in advance!
1. A 100-ft oil is to be drilled.The cost of drilling the first foot is $10.00, and the cost for drilling each additional foot is$.50 more thaan that of preceding foot.Find the cost of drilling the entire 100ft.
2.A tank contains 20 galloons of water.One half of the water is removed and replaced with antifreeze.This process is continued eight times.How much water remains in the tank after these eight replacements process?

2. Hello, slendersnake!

1. A 100-ft oil is to be drilled. The cost of drilling the first foot is $10.00. The cost for drilling each additional foot is$0.50 more than that of preceding foot.
Find the cost of drilling the entire 100 ft.
$\text{The total cost is: }\;C \;=\;\underbrace{10 + 10.5 + 11 + 11.5 + 12 + 12.5 + \hdots}_{\text{ 100 terms}}$

This is an Arithmetic Series with:
. . first term $a = 10$, common difference $d = \tfrac{1}{2}$, $n = 100$

Therefore: . $C \;=\;\tfrac{n}{2}[2a + (n-1)d] \;=\;\tfrac{100}{2}[2(10) + 99(\tfrac{1}{2})] \;=\;3475$

. . The cost is: . $\3,\!475$

2. A tank contains 20 gallons of water.
One half of the water is removed and replaced with antifreeze.
This process is continued eight times.
How much water remains in the tank after these eight replacements process?
After the 1st replacement, the amount of water is: . $\tfrac{1}{2}(20)$ gallons.

After the 2nd replacement, the amount of water is: . $(\tfrac{1}{2})^2(20)$ gallons.

After the 3rd replacement, the amount of water is: . $(\tfrac{1}{2})^3(20)$ gallons.

. . . . . etc. . . .

After the 8th replacement, the amount of water is: . $(\tfrac{1}{2})^8(20)$ gallons.

And: . $(\tfrac{1}{2})^8(20) \:=\:\frac{20}{256} \;=\;\boxed{\frac{5}{64}\text{ gallons}}$

3. thanks Soroban!
i have some confuse with problem 1
why the answer is not: C=a+(n-1)d=10+99(0.5)=59.5??