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Mathematics · 2023 · 3 marks
CBSE 2023 · Region 4 · Set 3 · Q29
How many terms are there in A.P. whose first and fifth term are - $\displaystyle 14$ and $\displaystyle 2$, respectively and the last term is 62.Which term of the A. P. : $\displaystyle 65,61,57,53$, is the first negative term?
How many terms are there in A.P. whose first and fifth term are - $\displaystyle 14$ and $\displaystyle 2$, respectively and the last term is 62.
Which term of the A. P. : $\displaystyle 65,61,57,53$, is the first negative term?
Marking-scheme solution
(A)
\[\begin{aligned}
& a=-14, a_{5}=2 \quad \Rightarrow a+4 d=2 \\
& \quad-14+4 d=2 \Rightarrow d=4 \\
& a_{n}=62 \quad \Rightarrow a+(n-1) d=62 \\
& -14+(n-1) 4=62 \quad \Rightarrow n=20
\end{aligned}
\]
(B)
$\displaystyle 65$, $\displaystyle 61$, $\displaystyle 57$, 53...
\[a=65, d=-4
\]
Let \(\displaystyle \mathrm{a}_{\mathrm{n}}\) be the first negative term
\[\begin{aligned}
& \mathrm{a}_{\mathrm{n}}<0 \Rightarrow \mathrm{a}+(\mathrm{n}-1) \mathrm{d}<0 \\
& 65+(\mathrm{n}-1)(-4)<0 \Rightarrow 69-4 \mathrm{n}<0 \\
& \quad \mathrm{n}>\frac{69}{4}
\end{aligned}
\]
∴ Least positive integral value of n which satisfies n > \(\displaystyle \frac{69}{4}\) is $\displaystyle 18$
\[\therefore 1^{\mathrm{St}} \text { negative term of } \mathrm{AP}=18
\]
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CBSE Class 10 Mathematics past-paper question from the 2023board exam, with the answer as CBSE’s own marking scheme gives it. Where our answers come from.