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The sum of first $q$ terms of an A.P. is $63q - 3q^2$. If its $p^{\text{th}}$ term is $-60$, find the value of $p$. Also, find the $11^{\text{th}}$ term of this A.P.
The sum of first $q$ terms of an A.P. is $63q - 3q^2$. If its $p^{\text{th}}$ term is $-60$, find the value of $p$. Also, find the $11^{\text{th}}$ term of this A.P.
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$S_q = 63q - 3q^2$
$\therefore S_1 = 60 \Rightarrow a_1 = 60$ ($1^{\text{st}}$ term)
$S_2 = 63(2) - 3(2)^2 = 126 - 12 = 114$
$a_1 + a_2 = 114 \Rightarrow a_2 = 114 - 60 = 54$
$d = a_2 - a_1 = 54 - 60 = -6$
$a_p = -60$
$60 + (p-1)d = -60$
$(p-1)(-6) = -120 \Rightarrow p=21$
$a_{11} = a + 10d = 60 + 10(-6) = 0$
$\therefore S_1 = 60 \Rightarrow a_1 = 60$ ($1^{\text{st}}$ term)
$S_2 = 63(2) - 3(2)^2 = 126 - 12 = 114$
$a_1 + a_2 = 114 \Rightarrow a_2 = 114 - 60 = 54$
$d = a_2 - a_1 = 54 - 60 = -6$
$a_p = -60$
$60 + (p-1)d = -60$
$(p-1)(-6) = -120 \Rightarrow p=21$
$a_{11} = a + 10d = 60 + 10(-6) = 0$