The nth term of a geometric sequence can be written
a(n) = a(1)*r^(n-1)
We can rewrite the two given equations:
a(2) + a(4) = 30 = a(1)*r + a(1)*r^3 (Equation 1)
a(3) + a(5) = 15 = a(1)*r^2 + a(1)*r^4 (Equation 2)
Notice that Equation 1 can be substituted into Equation, i.e.,
a(1)*r^2 + a(1)*r^4 = 15
r*[a(1)*r + a(1)*r^3] = 15
r*(30) = 15
r = 1/2
If we substitute this into Equation 1, we have
a(1)/2 + a(1)/8 = 30
a(1) = 30 * 8/5 = 48
The answer is 48. Did he even need to specify that the sum exists? It’s a finite geometric sequence so surely it’s impossible for the sum not to exist? If you want to stop me old man, ask me the most basic combinatorics question you can think of and I’ll start foaming at the mouth.
Thanks for verifying my answer, Elpam!
Sauce: The Last Adventurer, Ch. 103
It’s a sequence of numbers where the next number is a fixed multiple of the previous one.
1 2 4 8 16 32 … Is a sequence with a ratio of 2.
96 48 24 12 6 3 … Is the sequence from the question with a ratio of 1/2. You see that elements 2 and 4 (counting from zero) sum to 30 and elements 3 and 5 sum to 15.
In general, the sequence can be expressed as a_k = c r^k for some starting value c and the ratio r.