CAT 2023 Slot 1 QA Question 22: A lab experiment measures the number of organisms at 8 am every day. Starting with 2 organisms on the first day, the number of organisms on any day is equal to 3 more than twice the number on the prev

Question

A lab experiment measures the number of organisms at 8 am every day. Starting with 2 organisms on the first day, the number of organisms on any day is equal to 3 more than twice the number on the prev

Accepted Answer

Let the number of organisms on day n be $a_n$. Then

$a_1 = 2$
$a_n = 2a_{n-1} + 3$

Add 3 to both sides to make the recurrence geometric.
$a_n + 3 = 2(a_{n-1}+3)$

Since $a_1 + 3 = 5$, we get
$a_n + 3 = 5 	imes 2^{n-1}$
$a_n = 5 	imes 2^{n-1} - 3$

Now check powers around one million.
$a_{18} = 5