Consider all 3-term geometric sequences with first term 1 and with common ratio the square of an integer between 1 and 1000 (inclusive). How
many of these 1000 geometric sequences have the property that the sum of the 3 terms is prime?
To solve this problem, we need to find the number of geometric sequences that have the property of the sum of their terms being prime.
Let's analyze the problem step by step:
1. We are given that the first term of the geometric sequence is 1. So, the three terms will be 1, x, and x^2, where x is the common ratio of the sequence.
2. The sum of the three terms of the sequence is given by S = 1 + x + x^2.
3. We need to find the values of x for which the sum S is a prime number.
To find the number of sequences that satisfy this property, we need to check each value of x (the common ratio) between 1 and 1000 (inclusive) and determine if the sum S is prime.
Here's an algorithm to do that:
1. Initialize a counter variable to keep track of the number of sequences with prime sums. Set it to 0.
2. Iterate through each value of x from 1 to 1000 (inclusive).
3. Calculate the sum of the three terms using the formula S = 1 + x + x^2.
4. Check if the sum S is a prime number. To do this, check if S is divisible evenly by any number between 2 and the square root of S. If it is, then it is not a prime number. If it's not divisible by any number, then it is a prime number.
5. If the sum S is a prime number, increment the counter variable.
6. After iterating through all values of x, the counter variable will hold the number of sequences with prime sums.
Note: To efficiently check for prime numbers, you can use a primality testing algorithm like the Sieve of Eratosthenes or the Miller-Rabin primality test.
Implementing this algorithm in code will give you the desired answer.