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Q.
The number of terms in an $A.P$. is even; the sum of the odd terms in it is $24$ and that the even terms is $30$. If the last term exceeds the first term by $10 \frac{1}{2},$ then the number of terms in the $A.P$. is :
Let no. of terms $=2n$
$a ,( a + d ),( a +2 d )$, ,........ $a +(2 n -1) d$
sum of even terms
$\frac{ n }{2}[2( a + d )+( n -1) 2 d ]=30 \,\,\,\,\, ......(i)$
sum of odd terms
$\frac{ n }{2}[2 a +( n -1) 2 d ]=24\,\,\,\,\, .......(ii)$
$a +(2 n -1) d - a =\frac{21}{2}\,\,\,\,\, ......(iii)$
eq. (i)....eq. (ii)
$\frac{ n }{2} \times 2 d =6$
$ \Rightarrow nd =6 \,\,\,\, ......(iv)$
$(2 n -1) d =\frac{21}{2} \,\,\,\, ......(v)$
$\frac{ eq ( iv )}{ eq ( v )}=\frac{ n }{2 n -1}=\frac{4}{7}$
$ \Rightarrow 8 n -4=7 n $
$n =4$
so no. of terms $=8$