Integers a, b, c, d and e satisfy 50<a<b<c<d<e<500, and a,b,c,d,e form a geometric sequence. What is the sum of all possible distinct values of a?
500 = 50r^6
r^6 = 10
Since r is irrational, and no power of 5 less than 6 is rational, there are no integer values between 50 and 500.
@Steve: what about this sequence?
64,96,144,216,324
the answer is 321
To find the sum of all possible distinct values of a, we first need to find the possible values of a.
We are given that a, b, c, d, and e form a geometric sequence. In a geometric sequence, each term is found by multiplying the previous term by a constant factor called the common ratio.
Let's assume the common ratio is r. Then, we can write the terms of the geometric sequence as:
a, ar, ar^2, ar^3, ar^4
Since a, b, c, d, and e are integers, we need to find whole number solutions for a and r.
Since 50 < a < b < c < d < e < 500, we know that the largest possible value for a is 499, as a cannot be equal to or greater than b.
Now, we can analyze the range of possible values for r. Since d < e, we have ar^3 < ar^4, and since c < d, we have ar^2 < ar^3. Therefore, we have:
ar^2 < ar^3 < ar^4
To determine the smallest possible value for r, we need to find the largest value for r that satisfies ar^2 < ar^3 < ar^4. Let's consider the inequality ar^2 < ar^3:
Dividing both sides by ar^2, we get:
1 < r
This means that the common ratio must be greater than 1.
To determine the largest possible value for r, we need to find the smallest value for r that satisfies ar^3 < ar^4. Let's consider the inequality ar^3 < ar^4:
Dividing both sides by ar^3, we get:
1 < r
This shows that the common ratio must be greater than 1.
Therefore, the possible values for the common ratio r are between 1 and the largest integer smaller than (500/499), which is 1.
To find the possible values of a, we can use the possible values of r. Starting with a = 50 and using the possible values of r, we can calculate each term of the geometric sequence until we reach a value larger than 500.
Let's do the calculations:
For r = 2, a = 50, ar = 100, ar^2 = 200, ar^3 = 400, ar^4 = 800 (which is larger than 500)
For r = 3, a = 50, ar = 150, ar^2 = 450, ar^3 = 1350 (which is larger than 500)
For r = 4, a = 50, ar = 200, ar^2 = 800 (which is larger than 500)
We can see that for r = 2 and r = 3, the terms of the geometric sequence exceed 500, but for r = 4, the terms do not exceed 500.
Therefore, the possible values for a are 50, which occurs for r = 2, and 50, which occurs for r = 3.
The sum of all possible distinct values of a is 50 + 50 = 100.
So, the sum of all possible distinct values of a is 100.