Consider an A.P. of positive integers, whose sum of first three terms is 54 and sum of first twenty

English

Gujarati

Hindi

8. Sequences and Series

hard

Consider an $A.P.$ of positive integers, whose sum of the first three terms is $54$ and the sum of the first twenty terms lies between $1600$ and $1800$ . Then its $11^{\text {th }}$ term is :

A$84$

B$122$

C$90$

D$108$

(JEE MAIN-2025)

Solution

$S_3=3 a+3 d=54$
$\Rightarrow a+d=18$
$S_{20}=10(2 a+19 d)$
$\Rightarrow 10(36+17 d)$
$\Rightarrow 1600 < 10(36+17 d) < 1800$
$\Rightarrow 160 < 36+17 d < 180$
$\Rightarrow 124 < 17 d < 144$
$\Rightarrow 7 \frac{5}{17} < d < 8 \frac{8}{17}$
Common difference will be natural number
$\Rightarrow d=8 \Rightarrow a=10$
$\Rightarrow a_{11}=10+10 \times 8=90$

Standard 11

Mathematics

If in the equation $a{x^2} + bx + c = 0,$ the sum of roots is equal to sum of square of their reciprocals, then $\frac{c}{a},\frac{a}{b},\frac{b}{c}$ are in

hard

If sum of $n$ terms of an $A.P.$ is $3{n^2} + 5n$ and ${T_m} = 164$ then $m = $

easy

If the ${p^{th}},\;{q^{th}}$ and ${r^{th}}$ term of an arithmetic sequence are $a , b$ and $c$ respectively, then the value of $[a(q – r)$ + $b(r – p)$ $ + c(p – q)] = $

easy

Let $S_n$ and $s_n$ deontes the sum of first $n$ terms of two different $A.P$. for which $\frac{{{s_n}}}{{{S_n}}} = \frac{{3n – 13}}{{7n + 13}}$ then $\frac{{{s_n}}}{{{S_{2n}}}}$

normal

If ${a_1},\;{a_2},…………,{a_n}$ are in $A.P.$ with common difference , $d$, then the sum of the following series is $\sin d(\cos {\rm{ec}}\,{a_1}.co{\rm{sec}}\,{a_2} + {\rm{cosec}}\,{a_2}.{\rm{cosec}}\,{a_3} + ………..$$ + {\rm{cosec}}\;{a_{n – 1}}{\rm{cosec}}\;{a_n})$

medium