Question

log10,log100,log1000, is in which progression​

Answer 1

$Given \: sequence: \\ log \:10, \: log \:100, \: log \:1000$

$First \:term (a) = a_{1} = log \: 10$

$Second \:term (a_{2}) = log \: 100 \\= log \: 10^{2}\\= 2 log \: 10$

$\boxed { \pink { Since , log \:a^{m} = m log a }}$

$Third\:term (a_{3}) = log \: 1000 \\= log \: 10^{3} \\= 3 log \: 10$

$\red{ a_{2} - a_{1}} = 2log \:10 - log \:10\\= log \: 10\: ---(1)$

$\red{ a_{3} - a_{2}} = 3log \:10 - 2log \:10\\= log \: 10\: ---(2)$

/* From (1) and (2), */

$\red{ a_{2} - a_{1}} = \red{ a_{3} - a_{2}}$

$\blue {( Difference \: of \: consecutive \:terms }\\\blue { are \: equal ) }$

Therefore.,

$\green { Given \: terms \: are }\\\green {in \: Arithmetic \: progression }$

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