If $$a1$$,$$a2$$,…,an,… form a G.P. and ai $$0$$, for all i≥$$1$$, then log⁡an log⁡$$an+1$$ log⁡$$an+2log$$⁡$$an+3$$ log⁡$$an+4$$ log⁡$$an+5log$$⁡$$an+6$$ log⁡$$an+7$$ log⁡$$an+8$$ is equal to

Question

If $$a1$$,$$a2$$,…,an,… form a G.P. and ai > $$0$$, for all i≥$$1$$, then log⁡an       log⁡$$an+1$$    log⁡$$an+2log$$⁡$$an+3$$    log⁡$$an+4$$    log⁡$$an+5log$$⁡$$an+6$$    log⁡$$an+7$$    log⁡$$an+8$$ is equal to

Infinity Learn · Accepted Answer

We have,$$an+12=anan+2$$⇒ $$2log$$ ⁡$$an+1=log$$ ⁡$$an+log$$ ⁡$$an+2Similarly$$,$$2log$$⁡$$an+4=log$$⁡$$an+3+log$$⁡$$an+52log$$⁡$$an+7=log$$⁡$$an+6+log$$⁡$$an+8Substituting$$ these values in second column of determinant, we getΔ$$=12log$$⁡anlog⁡$$an+log$$⁡$$an+2log$$⁡$$an+2log$$⁡$$an+3log$$⁡$$an+3+log$$⁡$$an+5log$$⁡$$an+5log$$⁡$$an+6logn+6+log$$⁡$$an+8log$$⁡$$an+8=12(0)=0$$  [Using $$C2$$→$$C2$$−$$C1$$−$$C3$$

If $a_{1}, a_{2}, \dots, a_{n}, \dots$ form a G.P. and a_i > 0, for all $i \geq 1$ , then $|\begin{array}{lll} \log a_{n} \log a_{n + 1} \log a_{n + 2} \\ \log a_{n + 3} \log a_{n + 4} \log a_{n + 5} \\ \log a_{n + 6} \log a_{n + 7} \log a_{n + 8} \end{array}|$ is equal to

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If a1,a2,…,an,… form a G.P. and ai > 0, for all i≥1, then log⁡an log⁡an+1 log⁡an+2log⁡an+3 log⁡an+4 log⁡an+5log⁡an+6 log⁡an+7 log⁡an+8 is equal to

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If $a_{1}, a_{2}, \dots, a_{n}, \dots$ form a G.P. and a_i > 0, for all $i \geq 1$ , then $|\begin{array}{lll} \log a_{n} \log a_{n + 1} \log a_{n + 2} \\ \log a_{n + 3} \log a_{n + 4} \log a_{n + 5} \\ \log a_{n + 6} \log a_{n + 7} \log a_{n + 8} \end{array}|$ is equal to