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The sum of first $q$ terms of an A.P. is $63q – 3q^2$. If its $p$th term is $-60$, find the value of $p$, Also, find the 11th term of this A.P.
Given:
The sum of first $q$ terms of an A.P. is $63q – 3q^2$ and its $p$th term is $-60$.
To do:
We have to find the value of $p$ and $11^{th}$ term of the given A.P.
Solution:
$S_{q} =63q-3q^{2}$
For $q=1,\ S_{1} =63\times 1-3\times 1^{2}=63-3=60$
Therefore, first term $a=60$
For $n=2,\ S_{2} =63\times 2 - 3\times 2^2=126-12=114$
$\therefore$ Second term of the A.P.$=S_{2} -S_{1}$
$=114-60$
$=54$
Common difference of the A.P., $d=$second term $-$ first term
$=54-60=-6$
We know that,
$a_{q}=a+(q-1)d$
$a_{q}=60+( q-1) \times (-6)$
$-60=60-6q+6$
$6q=66+60$
$6q=126$
$q=21$
$a_{11}=60+( 11-1) \times (-6)$
$=60+10\times (-6)$
$=60-60$
$=0$
Therefore, the value of $q$ is $21$ and the $11^{th}$ term of the given A.P. is $0$. 
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