Let Ai be a sequence such that An/An-1 =p for all n. Let Bi be a sequence such that Bn/Bn-1 be q for all n>1
Further, let A1 = ¼ = p, B1 = ¾ = q.
Find the smallest n such that An + Bn < 0.1
The above sequences are nothing but Geometric Progressions. The first one has first term = common ratio = ¼. The second one has first term = common ratio = ¾
So, we are effectively solving for (¼)n + (¾)n < 0.01
Now, we can skip to trial and error. (¾)n will be far higher than (¼)n , so we can focus on that.
Let us take n = 4; (¾)n = 81/256. This is way higher than 0.01. This is closer to 1/3. If we square this, that will be close to 0.01.
So, we are looking at (¾)8 = 6561/65536 . This is just more than 0.1. We can be reasonably confident that n = 9 will work. The (¼)9 component gives us a minuscule number and can be ignored.
So, n =9 works best