A contest winner is given a choice of 2 prizes.
Prize 1:$1 today, $2 tomorrow, $4 on day 3 and so on, for 30 days. Each day the winner receives twice as much as the day before.
Prize 2: $1 today, $3 tomorrow, $9 on day 3, and so on, for 20 days. Each day the winner receives 3 times as much as the day before.
Which prize would you choose? Justify. Need the sum for each prize
compare the sums of the two geometric sequences:
(2^30 - 1)/(2-1)
vs
(3^20 - 1)/(3-1)
There's a difference, but either one will set you up for life!!
To determine which prize is better, we need to calculate the total amount received for each prize. Let's start with Prize 1 and calculate the sum of the amounts received over 30 days.
To simplify the calculation, let's represent the amount received on day 1 as 2^0, the amount received on day 2 as 2^1, the amount received on day 3 as 2^2, and so on. So the formula for the amount received on day n would be 2^(n-1).
Using this formula, we can find the amounts received for Prize 1 over 30 days:
On day 1: 2^0 = 1
On day 2: 2^1 = 2
On day 3: 2^2 = 4
And so on...
Using this pattern, we can calculate the sum of the amounts received for Prize 1 over 30 days:
Sum = 1 + 2 + 4 + 8 + ... + (2^29)
To get this sum, we can use the formula for the sum of a geometric series:
Sum = a * (r^n - 1) / (r - 1)
In this case, a (the first term) = 1, r (the common ratio) = 2, and n (the number of terms) = 30.
Plugging these values into the formula, we get:
Sum = 1 * (2^30 - 1) / (2 - 1)
= (2^30 - 1)
So the sum of the amounts received for Prize 1 is (2^30 - 1).
Now let's move on to Prize 2 and calculate the sum of the amounts received over 20 days using a similar approach.
On day 1: 3^0 = 1
On day 2: 3^1 = 3
On day 3: 3^2 = 9
And so on...
Using the pattern, the sum of the amounts received for Prize 2 over 20 days is:
Sum = 1 + 3 + 9 + 27 + ... + (3^19)
Using the formula for the sum of a geometric series, with a = 1, r = 3, and n = 20, we get:
Sum = 1 * (3^20 - 1) / (3 - 1)
= (3^20 - 1) / 2
So the sum of the amounts received for Prize 2 is (3^20 - 1) / 2.
Now that we have calculated the sums for both prizes, we can compare them to determine which prize is better.