Gujarat BoardEnglish MediumSTD 11 ScienceMATHSArithmetic Progressions3 Marks
Question
Solve:25 + 22 + 19 + 16 + ... + x = 115
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Answer
25 + 22 + 19 + 16 + ... + x = 115 Here, sum of the given series of say n terms is 115 So, the $n^{th}$ term = x Here, $\text{a}=25$ and $\text{d}=22-25=-3$ $\therefore\text{a}_\text{n}=\text{a}+(\text{n}-1)\text{d}$ $\Rightarrow\text{x}=25-3(\text{n}-1)$ $\Rightarrow\text{x}=28-3\text{n}\ ...(1)$ The sum of n terms $\text{s}_{\text{n}}=\frac{\text{n}}{2}[\text{a}+\text{l}]$ $\Rightarrow115=\frac{\text{n}}{2}[25+28-3\text{n}]$ $\Rightarrow230=53\text{n}-3\text{n}^2$ $\Rightarrow3\text{n}^2-53\text{n}-3\text{n}^2$ $\Rightarrow3\text{n}^2-30\text{n}-23\text{n}-230=0$ $\Rightarrow\text{n}=10$ or $\frac{23}{3}$ But n can't be function $\therefore\text{n}=10\ .....(2)$ From (1) and (2) $\text{x}=28-3\text{n}$ $=28-3(10)$ $=-2$ $\text{x}=-2$
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