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(C) Let $a$ be the first term and $d$ be the common difference of the $AP$.The $n^{	ext{th}}$ term of an $AP$ is given by $a_n = a + (n-1)d$.Accordin

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$3$

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(C) Let $a$ be the first term and $d$ be the common difference of the $AP$.The $n^{	ext{th}}$ term of an $AP$ is given by $a_n = a + (n-1)d$.According to the first condition,$a_8 = \frac{1}{2} a_2$.$a + 7d = \frac{1}{2}(a + d)$$2a + 14d = a + d$$a + 13d = 0$  --- $(i)$According to the second condition,$a_{11} = \frac{1}{3} a_4 + 1$.$a + 10d = \frac{1}{3}(a + 3d) + 1$$3a + 30d = a + 3d + 3$$2a + 27d = 3$  --- $(ii)$From equation $(i)$,$a = -13d$. Substituting this into equation $(ii)$:$2(-13d) + 27d = 3$$-26d + 27d = 3$$d = 3$Substituting $d = 3$ into equation $(i)$:$a + 13(3) = 0$$a = -39$Now,find the $15^{	ext{th}}$ term $(a_{15})$:$a_{15} = a + 14d$$a_{15} = -39 + 14(3)$$a_{15} = -39 + 42 = 3$.

The eighth term of an $AP$ is half its second term and the eleventh term exceeds one third of its fourth term by $1$. Find the $15^{\text{th}}$ term.

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