Problem Statement
An AP consists of $50$ terms of which $3^{\text{rd}}$ term is $12$ and the last term is $106$. Find the $29^{\text{th}}$ term.
Verified Solution & Marking Scheme
Set Up Equations for a₃ and a₅₀
$a_3 = a + 2d = 12$ ...(1)
$a_{50} = a + 49d = 106$ ...(2)
Subtract to Find Common Difference d and First Term a
$(a + 49d) - (a + 2d) = 106 - 12 \implies 47d = 94 \implies d = 2$
Substitute $d = 2$ in (1):
$a + 2(2) = 12 \implies a + 4 = 12 \implies a = 8$
Compute a₂₉
$a_{29} = a + 28d = 8 + 28(2) = 8 + 56 = 64$