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
Yeah, it makes no sense to specify that the sum exists; the finite sum always exists, and even if you were talking about the infinite sum, it makes no odds on what the first term is.
I suppose that this specification was said to make the task look more complicated to solve. Just a simple obstacle.