Download App

Arithmetic Progressions — Maths STD 10 — Question

Maharashtra BoardEnglish MediumSTD 10MathsArithmetic Progressions4 Marks
Question
Let there be an A.P. with first term $'a'$, common difference $'d'$. If $a_n$ denotes in $n^{th}$​​​​​​​ term and $S_n$​​​​​​​  the sum of first $n$ terms,
​​​​​​​find.$n$ and $a$, if $a_n = 4, d = 2$ and $S_n = -14.$
✓

Answer

$a_n=4, d=2, S_n=-14$
$\therefore a+(n-1) 2=4 \Rightarrow a+(n-1)^2=4$
$\Rightarrow a+2 n-2=4 \Rightarrow a+2 n=4+2$
$a+2 n=6 \ldots . . \text { (i) }$
$S_n=\frac{n}{2}[a+1] \Rightarrow-14=\frac{n}{2}[a+4]$
$n(a+4)=-28 \text {.....(ii) }$
From (i) $a=6-2 n$
$\therefore \text { in (ii) }$
$n(6-2 n+4)=-28 \Rightarrow n(10-2 n)=-28$
$\Rightarrow 10 n-2 n^2=-28 \Rightarrow 2 n^2-10 n-28=0$
$\left.\Rightarrow n^2-5 n-14=0 \text { (Divisible by } 2\right)$
$\Rightarrow n^2-7 n+2 n-14=0$
$\begin{Bmatrix} \therefore\ -14=-7\times2\\ -5=-7=2 \end{Bmatrix}$
$\Rightarrow n(n - 7) + 2(n - 7) = 0$
$\Rightarrow (n - 7)(n + 2) = 0$
Either $n-7=0$, then $n=7$
of $n +2=0$, then $n =-2$ but it is not possible being negative
$\therefore n =7$
Now $a =6-2 n =6-2 \times 7=6-14=-8$
Hence $a=-8, n=7$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "4 Marks Questions" from Arithmetic Progressions→Arithmetic Progressions — all question types→Maths STD 10 question papers→Maharashtra Board STD 10 question papers→Maharashtra Board question paper generator→

Similar questions

Evaluate the following:
$\frac{4}{\cot^230^\circ}+\frac{1}{\sin^260^\circ}-\cos^245^\circ$
→
In the given figure, a triangle ABC is drawn to circumscribe a circle of radius 2cm such that the segments BD and DC into which BC is divided by the point of contact D, are of lengths 4cm and 3cm respectively. If the area of $\triangle\text{ABC} = 21\text{cm}^2$ then find the lengths of sides of AB and AC.
→
Find the value of k for which the root are real and equal in the following equations:
$(2k + 1)x^2 + 2(k + 3)x + (k + 5) = 0$
→
Find the length of the side and perimeter of an equilateral triangle whose height is $\sqrt3$ cm.
→
Prove that the points (3, -2), (4, 0), (6, -3) and (5, -5) are the vertices of a parallelogram.
→
A person deposits ₹ $x$ in savings bank account at the rate of $5 \%$ per annum and ₹ y in fixed deposit at $10 \%$ per annum. At the end of one year, he gets ₹ 400 as total interest. If he deposits ₹ y in savings bank account and ₹ $x$ in fixed deposit, he would get ₹ 350 as total interest. Find the total amount he deposited.
→
The frequency distribution table shows the number of mango trees in a grove and their yield of mangoes. Find the median of data.
→
To fill a swimming pool two pipes are used. If the pipe of larger diameter used for $4$ hours and the pipe of smaller diameter for $9$ hours, only half of the pool can be filled. Find, how long it would take for each pipe to fill the pool separately, if the pipe of smaller diameter takes $10$ hours more than the pipe of larger diameter to fill the pool?
→
Using the distance formula, show taht the given points are collinear:
(-2, 5), (0, 1) and (2, -3).
→
To find the width of the river, a man observes the top of a tower on the opposite bank making an angle of elevation of $61^{\circ}$. When he moves $50 \mathrm{~m}$ backword from bank and observes the same top of the tower, his line of vision makes an angle of elevation of $35^{\circ}$. Find the height of the tower and width of the river. $\left(\tan 61^{\circ}=1.8, \tan 35^{\circ}=0.7\right)$
→