Question Number 22726 by math solver last updated on 22/Oct/17
Commented by math solver last updated on 24/Oct/17
$$\:{solve}\:{q}.\mathrm{8}? \\ $$
Commented by math solver last updated on 24/Oct/17
$$\left.{at}\:{last}\:{i}\:{was}\:{able}\:{to}\:{do}\:{q}.\mathrm{8}\:{also}\::\right) \\ $$
Answered by $@ty@m last updated on 22/Oct/17
$${Solution}\:{of}\:\mathrm{9}. \\ $$$$\mathrm{1}+\mathrm{2}+\mathrm{4}+\mathrm{8}+….+\mathrm{512}\:\:\:{is}\:{in}\:{GP} \\ $$$$\therefore\:{t}_{{n}} =\mathrm{2}^{{n}−\mathrm{1}} \:−−\left(\mathrm{1}\right) \\ $$$$\mathrm{1}+\mathrm{4}+\mathrm{7}+….\:+\mathrm{298}\:{is}\:{in}\:{AP} \\ $$$$\therefore\:{t}_{{m}} =\mathrm{1}+\left({m}−\mathrm{1}\right).\mathrm{3} \\ $$$$\Rightarrow{t}_{{m}} =\mathrm{3}{m}−\mathrm{2}\:−−−\left(\mathrm{2}\right) \\ $$$${We}\:{have}\:{to}\:{find}\:{that}\:{for} \\ $$$${how}\:{many}\:{n},{m}\in{N} \\ $$$$\left(\mathrm{1}\right)\equiv\left(\mathrm{2}\right) \\ $$$$\Rightarrow\mathrm{2}^{{n}−\mathrm{1}} \equiv\mathrm{3}{m}−\mathrm{2} \\ $$$$\Rightarrow\mathrm{2}^{{n}−\mathrm{1}} +\mathrm{2}\equiv\mathrm{3}{m} \\ $$$$\Rightarrow\mathrm{2}^{{n}−\mathrm{1}} +\mathrm{2}\:{is}\:{a}\:{multiple}\:{of}\:\mathrm{3} \\ $$$$\Rightarrow{n}\in\left\{\mathrm{2},\mathrm{3},\:\mathrm{5},\mathrm{7},\:\mathrm{9}\right\} \\ $$$$\Rightarrow\:{Only}\:\mathrm{5}\:{terms}\:\mathrm{1},\:\mathrm{4}\:,\mathrm{16}\:,\:\mathrm{64}\:\&\:\mathrm{256}\:{are}\:{common} \\ $$$${in}\:{them}. \\ $$
Commented by math solver last updated on 22/Oct/17
$${wrong}\:{answer}!\:{the}\:{correct}\:{ans}.\:{is}\:\mathrm{5} \\ $$
Commented by math solver last updated on 22/Oct/17
$$\left.{don}'{t}\:{you}\:{think}\:\mathrm{1}\:{is}\:{also}\:{common}\::\right) \\ $$$$\:{the}\:{common}\:{terms}\:{are}\: \\ $$$$ \\ $$$$\mathrm{1},\mathrm{4},\mathrm{16},\mathrm{64},\mathrm{256}! \\ $$
Commented by $@ty@m last updated on 22/Oct/17
$${Thanks}\:{for}\:{pointing}\:{out}\:{my}\:{mistake}. \\ $$
Commented by math solver last updated on 22/Oct/17
$${no}\:{problem} \\ $$