Question Number 208218 by hardmath last updated on 07/Jun/24
$$\mathrm{a}_{\boldsymbol{\mathrm{n}}} \:\:\mathrm{numbers}\:\mathrm{series} \\ $$$$\mathrm{If}\:\:\mathrm{S}_{\mathrm{16}} \:−\:\mathrm{S}_{\mathrm{13}} \:\:=\:\:\mathrm{S}_{\mathrm{106}} \:−\:\mathrm{S}_{\mathrm{103}} \\ $$$$\mathrm{Find}:\:\:\:\:\frac{\mathrm{3a}_{\mathrm{3}} \:+\:\mathrm{4a}_{\mathrm{4}} \:+\:\mathrm{5a}_{\mathrm{5}} }{\mathrm{2a}_{\mathrm{12}} }\:\:=\:\:? \\ $$
Commented by mr W last updated on 07/Jun/24
$${this}\:{is}\:{not}\:{the}\:{first}\:{time}\:{when}\:{i}\:{ask} \\ $$$${you}\:{what}\:{you}\:{mean}\:{with}\:“{numbers} \\ $$$${series}'',\:{e}.{g}.\:{in}\:{Q}\mathrm{207930}. \\ $$
Commented by hardmath last updated on 07/Jun/24
$$ \\ $$For example, there is a numerical series and a geometric series
Commented by mr W last updated on 08/Jun/24
$${then}\:{you}\:{mean}\:{arithmetic}\:{series}. \\ $$
Commented by hardmath last updated on 08/Jun/24
$$\mathrm{yes}\:\mathrm{dear}\:\mathrm{professor} \\ $$
Answered by zamin2001 last updated on 07/Jun/24
$${S}_{\mathrm{16}} −{S}_{\mathrm{13}} ={a}_{\mathrm{16}} +{a}_{\mathrm{15}} +{a}_{\mathrm{14}} =\mathrm{3}{a}_{\mathrm{1}} +\mathrm{42}{d}\:\bigstar \\ $$$${S}_{\mathrm{106}} −{S}_{\mathrm{103}} ={a}_{\mathrm{106}} +{a}_{\mathrm{105}} +{a}_{\mathrm{104}} =\mathrm{3}{a}_{\mathrm{1}} +\mathrm{312}{d}\blacksquare \\ $$$$\:\bigstar\:=\blacksquare\:\Rightarrow\:{d}=\mathrm{0}\:\Rightarrow\:{constant}\:{numerical}\:{series} \\ $$$$\:\:\frac{\mathrm{3a}_{\mathrm{3}} \:+\:\mathrm{4a}_{\mathrm{4}} \:+\:\mathrm{5a}_{\mathrm{5}} }{\mathrm{2a}_{\mathrm{12}} }\:\:=\:\:\frac{\mathrm{3}{C}+\mathrm{4}{C}+\mathrm{5}{C}}{\mathrm{2}{C}}=\frac{\mathrm{12}{C}}{\mathrm{2}{C}}=\mathrm{6} \\ $$