Step-by-Step Solution
Key Concept: Main: AP $12, 15, \dots, 99$ ($n=30$). OR: $a_n = S_n - S_{n-1}$.
[Main Question Solution]
AP: $12, 15, \dots, 99$. $a=12, d=3, l=99 \Rightarrow 99 = 12 + (n-1)3 \Rightarrow n = 30$. [1.5 Marks]
$S_{30} = (30/2)(12 + 99) = 15 \times 111 = 1665$. [1.5 Marks]
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[OR Choice Question Solution]
$a_n = S_n - S_{n-1} = (3n^2 + 5n) - (3(n-1)^2 + 5(n-1)) = 6n + 2$. [2.0 Marks]
$a_{15} = 6(15) + 2 = 92$. [1.0 Mark]
Correct Answer: Main: 1665 | OR: a_n = 6n + 2, a_15 = 92