Question

How many terms will it take to get the sum $1046.5$ in AP $7$, $10\frac{1}{2}$, $14$,......

Answer 1

first term of AP$=a=7$

common difference =$10\frac{1}{2}-7=3\frac{1}{2}$

Sum of AP of $n$ terms is$\frac{n}{2}[2a+(n-1)d]$

so $\frac{n}{2}[2a+(n-1)d]$$=1046.5$

$\Rightarrow \frac{n}{2}[2\times 7+(n-1)3.5]=1046.5\Rightarrow 14n+3.5n^2-3.5n=1046.5\times 2\Rightarrow 3,5n^2+10,5n=2093\Rightarrow n^2+3n-598=0\Rightarrow n^2+26n-23n-598=0\Rightarrow n(n+26)-23(n+26)=0\Rightarrow (n-23)(n+26)=0\Rightarrow n=23$

as $n$ is a positive number

so number of terms is $23$