Complexity

The agent group meets regularly to discuss how the assignments have gone. The agents $26$ and $3$ insist on going through all previous meetings in detail at each meeting. Agent$-$ the manager notes that this leads to each meeting taking $3$ times as long as the previous one, plus $12$ minutes for coffee.

a$)$ The very first meeting lasted $48$ minutes. Set up a formula for a recursive sequence of numbers $\{$an$\}$

$1$

where an is the length of session number n$.$

b$)$ Using your formula, calculate the length of the first five sessions. Agent $11$ concludes that the meeting length can be written with the formula an $=183$ n $6.$ c$)$ Prove that your recursive formula matches this using induction.

$($The end result of the calculation was that the agent manager told agents $26$ and $3$ to shut up when they wanted to hold that briefing.$)$