**Question**

Let A_{i} be a sequence such that A_{n}/A_{n-1} =p for all n. Let B_{i} be a sequence such that B_{n}/B_{n-1} be q for all n>1

Further, let A_{1} = ¼ = p, B_{1} = ¾ = q.

Find the smallest n such that A_{n} + B_{n} < 0.1

**Solution **

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