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.
$a,$ if $a_n= 28, S_n= 144$ and $n = 9.$
$a,$ if $a_n= 28, S_n= 144$ and $n = 9.$
✓
Answer
$ a_n=28, S_n=144, n=9 $
$ a_n=a+(n-1) d $
$ \Rightarrow a_9=a+(9-1) d=a+8 d$
$a + 8d = 28 .....(i)$
$\text{S}_\text{n}=\frac{\text{n}}{2}[\text{a}+\text{l}]\Rightarrow\ \frac{9}{2}(\text{a}+28)=144$
$\Rightarrow\ \text{a}+28=\frac{144\times2}{9}=32$
$\text{a}=32-48=4$
$\therefore$ From $(i)$
$4+8\text{d}=28\Rightarrow\ 8\text{d}=28-4=24$
$\Rightarrow\ \text{a}=\frac{24}{8}=3$
Hence $a = 4.$
$ a_n=a+(n-1) d $
$ \Rightarrow a_9=a+(9-1) d=a+8 d$
$a + 8d = 28 .....(i)$
$\text{S}_\text{n}=\frac{\text{n}}{2}[\text{a}+\text{l}]\Rightarrow\ \frac{9}{2}(\text{a}+28)=144$
$\Rightarrow\ \text{a}+28=\frac{144\times2}{9}=32$
$\text{a}=32-48=4$
$\therefore$ From $(i)$
$4+8\text{d}=28\Rightarrow\ 8\text{d}=28-4=24$
$\Rightarrow\ \text{a}=\frac{24}{8}=3$
Hence $a = 4.$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
Solve the following quadratic equation:
$4\sqrt6\text{x}^2-\text{13x}-2\sqrt6=0$
→$4\sqrt6\text{x}^2-\text{13x}-2\sqrt6=0$
The sum of the $4th$ and $8th$ terms of an $AP$ is $24$ and the sum of the $6th$ and $10th$ term is $44$. Find the first three terms of the $AP.$
→Find the value of a and b for which the following systems of linear equations has an infinite number of solutions:
$2x + 3y = 7,$
$(a + b)x + (2a - b)y = 21$
→$2x + 3y = 7,$
$(a + b)x + (2a - b)y = 21$
Find the zeros of the polynomial $x^2+2 x-195$
→Find the mean of each the following frequency distributions: $(5 – 14)$
→| Class interval | $0-6$ | $6-12$ | $12-18$ | $18-24$ | $24-30$ |
| Frequency | $6$ | $8$ | $10$ | $9$ | $7$ |
In the given figure, $\angle\text{AMN}=\angle\text{MBC}=76^\circ.$ If $p, q$ and $r$ are the lengths of $AM, MB$ and $BC$ respectively then express the length of $MN$ in terms of $p, q$ and $r.$
→If $\cot\theta=\frac{1}{\sqrt{3}},$ write the value of $\frac{1-\cos^2\theta}{2-\sin^2\theta}.$
→A vertical tower stands on a horizontal plane and is surmounted by a vertical flag-staff. At a point on the plane $70$ metres away from the tower, an observer notices that the angles of elevation of the top and the bottom of the flagstaff are respectively $60^\circ $ and $45^\circ $. Find the height of the flag-staff and that of the tower.
→Evaluate the following:
$\sin^230^\circ\cos^245^\circ+4\tan^230^\circ+\frac{1}{2}\sin^290^\circ-2\cos^290^\circ+\frac{1}{24}\cos^20^\circ$
→$\sin^230^\circ\cos^245^\circ+4\tan^230^\circ+\frac{1}{2}\sin^290^\circ-2\cos^290^\circ+\frac{1}{24}\cos^20^\circ$
Write time first five terms of the following sequances whose $n^{th}$ terms are:
$a_n= 3n + 2$
→$a_n= 3n + 2$