If {a_1}, {a_2}, {a_3}, dots, {a_n}, dots are | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) Given that ${a_1}, {a_2}, {a_3}, \dots$ are in $G.P.$ with common ratio $r$.Thus,${a_{n+1}} = {a_n} \cdot r$,which implies $\log {a_{n+1}} = \log

Vedclass · Accepted Answer

$0$

Vedclass · Answer

(D) Given that ${a_1}, {a_2}, {a_3}, \dots$ are in $G.P.$ with common ratio $r$.Thus,${a_{n+1}} = {a_n} \cdot r$,which implies $\log {a_{n+1}} = \log {a_n} + \log r$.Similarly,$\log {a_{n+k}} = \log {a_n} + k \log r$.Let the determinant be $\Delta = \left| \begin{array}{ccc} \log {a_n} & \log {a_{n+1}} & \log {a_{n+2}} \ \log {a_{n+3}} & \log {a_{n+4}} & \log {a_{n+5}} \ \log {a_{n+6}} & \log {a_{n+7}} & \log {a_{n+8}} \end{array} ight|$.Applying column operations ${C_2} 	o {C_2} - {C_1}$ and ${C_3} 	o {C_3} - {C_2}$:$\Delta = \left| \begin{array}{ccc} \log {a_n} & \log {a_{n+1}} - \log {a_n} & \log {a_{n+2}} - \log {a_{n+1}} \ \log {a_{n+3}} & \log {a_{n+4}} - \log {a_{n+3}} & \log {a_{n+5}} - \log {a_{n+4}} \ \log {a_{n+6}} & \log {a_{n+7}} - \log {a_{n+6}} & \log {a_{n+8}} - \log {a_{n+7}} \end{array} ight|$.Since $\log {a_{n+k}} - \log {a_{n+k-1}} = \log r$ for any $k$,the determinant becomes:$\Delta = \left| \begin{array}{ccc} \log {a_n} & \log r & \log r \ \log {a_{n+3}} & \log r & \log r \ \log {a_{n+6}} & \log r & \log r \end{array} ight|$.Since column $2$ and column $3$ are identical,the value of the determinant is $0$.

If ${a_1}, {a_2}, {a_3}, \dots, {a_n}, \dots$ are in $G.P.$,then the value of the determinant $\left| \begin{array}{ccc} \log {a_n} & \log {a_{n+1}} & \log {a_{n+2}} \\ \log {a_{n+3}} & \log {a_{n+4}} & \log {a_{n+5}} \\ \log {a_{n+6}} & \log {a_{n+7}} & \log {a_{n+8}} \end{array} \right|$ is

Similar Questions

If $x, y, z$ are distinct and $\Delta=\left|\begin{array}{lll}x & x^{2} & 1+x^{3} \\ y & y^{2} & 1+y^{3} \\ z & z^{2} & 1+z^{3}\end{array}\right|=0,$ then show that $1+x y z=0$.

Using the property of determinants and without expanding,prove that:
$\left|\begin{array}{lll}2 & 7 & 65 \\ 3 & 8 & 75 \\ 5 & 9 & 86\end{array}\right|=0$

Evaluate $\Delta=\left|\begin{array}{lll}3 & 2 & 3 \\ 2 & 2 & 3 \\ 3 & 2 & 3\end{array}\right|$

If $\left|\begin{array}{ccc}a+b+2c & a & b \\ c & 2a+b+c & b \\ c & a & a+2b+c\end{array}\right|=2$,then $a^3+b^3+c^3-3abc=$

By using properties of determinants,show that:
$\left|\begin{array}{ccc}a^{2}+1 & a b & a c \\ a b & b^{2}+1 & b c \\ c a & c b & c^{2}+1\end{array}\right|=1+a^{2}+b^{2}+c^{2}$

Vedclass Test Series

Exam Paper Generator

Online Exam Module