CAT Online Coaching – Interesting question from Progressions

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 A₁ = ¼ = p, B₁ = 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