The first prize in a lottery is $250 000. Each winner chosen after the first is paid 20% as much as the winner before them. Determine a formula for the sum of n terms of the geometric sequence.
Clearly a = 250000
r = 1/5
Now just use your sum formula.
To find the sum of the first n terms of a geometric sequence, we can use the formula:
S = a * (1 - r^n) / (1 - r),
Where S represents the sum of the first n terms, a is the first term, r is the common ratio, and n is the number of terms.
In this case, the first term (a) is $250,000, and the common ratio (r) is 0.2 (since each winner is paid 20% as much as the previous winner). Let's substitute these values into the formula:
S = 250,000 * (1 - 0.2^n) / (1 - 0.2).
Simplifying further, we have:
S = 250,000 * (0.8^n - 1) / 0.8.
Therefore, the formula for the sum of n terms of the geometric sequence is:
S = 250,000 * (0.8^n - 1) / 0.8.