Two arithmetic sequences, with a twist

Algebra Level 4

Given that we have two arithmetic sequences \(\{a_n\}, \{b_n\}\) where

\(a_1 = 9\), \(a_{n+1} - a_n = 3\) for all natural \(n\)

\(b_1 = 0\), \(b_{n+1} - b_n = 5\) for all natural \(n\)

Now, find the smallest possible sum of the first \(x\) terms of the sequence \(\{a_n\}\) and the first \(y\) terms of the sequence \(\{b_n\}\) given that \(x+y = 10\), and \(x,y\) are natural numbers.

\(\color{white}{Last time I hid a secret message in one of my problems, only Rishabh found it. How about this one? Comment when you see it!}\)


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