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Question
Which term of the sequence 24, \[23\frac{1}{4,} 22\frac{1}{2,} 21\frac{3}{4}\]....... is the first negative term?
Solution
24,
\[23\frac{1}{4,}22\frac{1}{2,}21\frac{3}{4}\]
This is an A.P.
Here, we have:
a = 24
\[d = \left( 23\frac{1}{4} - 24 \right) = $\left( - \frac{3}{4} \right)$\]
\[\text { Let the first negative term be } a_n . \]
\[\text { Then, we have }: \]
\[ a_n < 0\]
\[ \Rightarrow a + \left( n - 1 \right) d < 0\]
\[ \Rightarrow 24 + \left( n - 1 \right) \left( - \frac{3}{4} \right) < 0\]
\[ \Rightarrow 24 - \frac{3n}{4} + \frac{3}{4} < 0\]
\[ \Rightarrow 24 + \frac{3}{4} < \frac{3n}{4}\]
\[ \Rightarrow \frac{99}{4} < \frac{3n}{4}\]
\[ \Rightarrow 99 < 3n\]
\[ \Rightarrow n > 33\]
Thus, the 34th term is the first negative term of the given A.P.
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