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Mathematics · 2024 · 3 marks
CBSE 2024 · Region 5 · Set 1 · Q27
(a) If the sum of the first 14 terms of an A.P. is 1050 and the first term is 10, then find the 20^th term and the n^th term.
(a)
If the sum of the first $\displaystyle 14$ terms of an A.P. is $\displaystyle 1050$ and the first term is $\displaystyle 10$ , then find the $\displaystyle 20^{\text {th }}$ term and the $\displaystyle \mathrm{n}^{\text {th }}$ term.
OR
(b)
The first term of an A.P. is $\displaystyle 5$, the last term is $\displaystyle 45$ and the sum of all the terms is 400. Find the number of terms and the common difference of the A.P.
Marking-scheme solution
(a)
\(\displaystyle \frac{14}{2}(20+13 d)=1050\)
\(\displaystyle \Rightarrow \mathrm{d}=10\)
\(\displaystyle \therefore \mathrm{a}_{20}=10+19 \times 10=200\)
\(\displaystyle \mathrm{a}_{\mathrm{n}}=10+(\mathrm{n}-1) 10=10 \mathrm{n}\)
OR
(b)
\(\displaystyle \mathrm{a}=5, \mathrm{a}_{\mathrm{n}}=45, \mathrm{~S}_{\mathrm{n}}=400\)
\[\begin{aligned}
& \frac{n}{2}(5+45)=400 \\
\Rightarrow & \mathrm{n}=16 \\
& 5+15 \mathrm{~d}=45 \\
\Rightarrow & \mathrm{~d}=\frac{40}{15} \text { or } \frac{8}{3}
\end{aligned}
\]
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