If 3^{2 sin 2 α-1},14 and 3^{4-2 sin 2 α} are first three terms of an A.P. for some α , then sixt

English

Gujarati

Hindi

8. Sequences and Series

hard

If $3^{2 \sin 2 \alpha-1},14$ and $3^{4-2 \sin 2 \alpha}$ are the first three terms of an $A.P.$ for some $\alpha$, then the sixth term of this $A.P.$ is

$66$

$65$

$81$

$78$

(JEE MAIN-2020)

Solution

Given that

$3^{4-\sin 2 \alpha}+3^{2 \sin 2 \alpha-1}=28$

Let $3^{2} \sin 2 \alpha=t$

$\frac{81}{t}+\frac{t}{3}=28$

$t=81,3$

$3^{2 \sin 2 \alpha}=3^{1}, 3^{4}$

$2 \sin 2 \alpha=1,4$

$\sin 2 \alpha=\frac{1}{2}, 2($ rejected $)$

First term $a=3^{2} \sin 2 \alpha-1$

$a=1$

Second term $=14$

$\therefore$ common difference $d=13$

$T_{6}=a+5 d$

$T _{6}=1+5 \times 13$

$T_{6}=66$

Standard 11

Mathematics

The arithmetic mean of the nine numbers in the given set $\{9,99,999,…., 999999999\}$ is a $9$ digit number $N$, all whose digits are distinct. The number $N$ does not contain the digit

normal

If ${a^2},\,{b^2},\,{c^2}$ be in $A.P.$, then $\frac{a}{{b + c}},\,\frac{b}{{c + a}},\,\frac{c}{{a + b}}$ will be in

medium

For three positive integers $p , q , r , x ^{ pq p ^2}= y ^{ qr }= z ^{ p ^2 r }$ and $r=p q+1$ such that $3,3 \log _y x, 3 \log _z y, 7 \log _x z$ are in A.P. with common difference $\frac{1}{2}$. Then $r – p – q$ is equal to

hard

(JEE MAIN-2023)

If $x,y,z$ are in $A.P.$ and ${\tan ^{ – 1}}x,{\tan ^{ – 1}}y$ and ${\tan ^{ – 1}}z$ are also in other $A.P.$ then . . .

medium

(JEE MAIN-2013)

The sums of $n$ terms of two arithmatic series are in the ratio $2n + 3:6n + 5$, then the ratio of their ${13^{th}}$ terms is

medium

If $3^{2 \sin 2 \alpha-1},14$ and $3^{4-2 \sin 2 \alpha}$ are the first three terms of an $A.P.$ for some $\alpha$, then the sixth term of this $A.P.$ is

Solution

Similar Questions

The arithmetic mean of the nine numbers in the given set $\{9,99,999,…., 999999999\}$ is a $9$ digit number $N$, all whose digits are distinct. The number $N$ does not contain the digit

If ${a^2},\,{b^2},\,{c^2}$ be in $A.P.$, then $\frac{a}{{b + c}},\,\frac{b}{{c + a}},\,\frac{c}{{a + b}}$ will be in

For three positive integers $p , q , r , x ^{ pq p ^2}= y ^{ qr }= z ^{ p ^2 r }$ and $r=p q+1$ such that $3,3 \log _y x, 3 \log _z y, 7 \log _x z$ are in A.P. with common difference $\frac{1}{2}$. Then $r – p – q$ is equal to

If $x,y,z$ are in $A.P.$ and ${\tan ^{ – 1}}x,{\tan ^{ – 1}}y$ and ${\tan ^{ – 1}}z$ are also in other $A.P.$ then . . .

The sums of $n$ terms of two arithmatic series are in the ratio $2n + 3:6n + 5$, then the ratio of their ${13^{th}}$ terms is

Start a Free Trial Now