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If the average of the first $n$ numbers in the sequence $148,146,144, \ldots$, is 125 , then $n$ is equal to
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Verified Answer
The correct answer is:
24
Given sequence $148,146,144, \ldots$ and average is 125.
$$
\begin{aligned}
\therefore \quad S_n & =\frac{n}{2}[2 \times 148+(n-1)(-2)] \quad[\because d=-2] \\
& =\frac{n}{2}[296-2 n+2] \\
& =n[148-n+1]=n[149-n]
\end{aligned}
$$
Hence, $\quad 125=\frac{n[149-n]}{n}$
$$
\begin{aligned}
& \Rightarrow \quad 125=149-n \\
& \therefore \quad n=149-125=24 \\
&
\end{aligned}
$$
$$
\begin{aligned}
\therefore \quad S_n & =\frac{n}{2}[2 \times 148+(n-1)(-2)] \quad[\because d=-2] \\
& =\frac{n}{2}[296-2 n+2] \\
& =n[148-n+1]=n[149-n]
\end{aligned}
$$
Hence, $\quad 125=\frac{n[149-n]}{n}$
$$
\begin{aligned}
& \Rightarrow \quad 125=149-n \\
& \therefore \quad n=149-125=24 \\
&
\end{aligned}
$$
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