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Arithmetic Progressions — MATHS STD 11 Science — Question

Rajasthan BoardEnglish MediumSTD 11 ScienceMATHSArithmetic Progressions3 Marks
Question
Solve:

25 + 22 + 19 + 16 + ... + x = 115

✓

Answer

25 + 22 + 19 + 16 + ... + x = 115
Here, sum of the given series of say n terms is 115
So, the nth 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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