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Question Number 114511 by 1549442205PVT last updated on 19/Sep/20

$$\mathrm{Find}\mathrm{numbers}\mathrm{which}\mathrm{are}\mathrm{common}$$$$\mathrm{terms}\mathrm{of}\mathrm{the}\mathrm{two}\mathrm{following}\mathrm{arithmetic}$$$$\mathrm{progression}:$$$$3,7,11,...,407\mathrm{and}2,9,16,...,709$$

Commented by bobhans last updated on 19/Sep/20

$$let{A}_n=\{3,7,11,...,23,...,51,...,407\}$$$$B_n=\{2,9,16,23,...,51,...,709\}$$$$wewantfind{C}_n=A_n\cap B_n$$$$consider{C}_n=\{23,51,...\}$$$$C_n=23+(n-1).28$$$$C_n=28n-5<407$$$$28n<412;n=14$$$$therefore{C}_n=\{23,51,79,...,387\}$$

Commented by bemath last updated on 19/Sep/20

$$superb...gavekudos$$

Commented by 1549442205PVT last updated on 19/Sep/20

$$\mathrm{Thank}\mathrm{Sir}.$$

Answered by PRITHWISH SEN 2 last updated on 19/Sep/20

$$3+({\mathrm{n}}_1-1)4=2+({\mathrm{n}}_2-1)7$$$${4\mathrm{n}}_1-1={7\mathrm{n}}_2-5$$$${7\mathrm{n}}_2-{4\mathrm{n}}_1=4$$$${7\mathrm{n}}_2=4({\mathrm{n}}_1+1)$$$$\because 4\mid {\mathrm{n}}_2\mathbf{and}7\mid ({\mathbf{n}}_1+1)$$$$\Rightarrow {\mathrm{n}}_1=6\mathrm{and}{\mathrm{n}}_2=4$$$$\therefore \mathrm{the}{1}^{\mathrm{st}}\mathrm{term}\mathrm{is}=4.6-1=7.4-5=23$$$$\mathrm{Now}\mathrm{we}\mathrm{have}\mathrm{to}\mathrm{find}\mathrm{an}\mathrm{A}.\mathrm{P}\mathrm{whose}{1}^{\mathrm{st}}\mathrm{term}\mathrm{is}23$$$$\mathrm{and}\mathrm{c}.\mathrm{d}\mathrm{is}=4.7=28\mathrm{and}\mathrm{last}\mathrm{term}\mathrm{is}⩽407$$$$\therefore {\mathbf{t}}_{\mathbf{n}}=23+(\mathbf{n}-1)28⩽407$$$$\Rightarrow (\mathbf{n}-1)28⩽384\Rightarrow \mathrm{n}⩽14$$$$\therefore \mathrm{the}\mathrm{no}.\mathrm{of}\mathrm{terms}=14$$$$$$$$$$

Commented by Rasheed.Sindhi last updated on 19/Sep/20

$$\mathcal{N}iceci\mathcal{N}$$$$\mathcal{S}iri\mathcal{S}$$$$!$$

Commented by PRITHWISH SEN 2 last updated on 19/Sep/20

$$\mathrm{thank}\mathrm{you}\mathrm{sir}.$$

Commented by 1549442205PVT last updated on 19/Sep/20

$$\mathrm{Thank}\mathrm{sir},\mathrm{solution}\mathrm{is}\mathrm{pretty}\mathrm{clear}$$

Answered by 1549442205PVT last updated on 19/Sep/20

$$\mathrm{It}\mathrm{is}\mathrm{clear}\mathrm{that}\mathrm{the}\mathrm{general}\mathrm{term}\mathrm{of}\mathrm{the}$$$$\mathrm{first}\mathrm{A}.\mathrm{P}\mathrm{is}\mathrm{of}\mathrm{the}\mathrm{form}{\mathrm{a}}_{\mathrm{n}}=3+4(\mathrm{n}-1);$$$$\mathrm{the}\mathrm{indicated}\mathrm{terms}\mathrm{of}\mathrm{the}\mathrm{progression}$$$$\mathrm{are}\mathrm{associated}\mathrm{with}\mathrm{the}\mathrm{values}\mathrm{n}=1,2,...$$$$,102.\mathrm{Similarly},\mathrm{the}\mathrm{terms}\mathrm{of}\mathrm{the}\mathrm{second}$$$$\mathrm{A}.\mathrm{P}\mathrm{are}\mathrm{obtained}\mathrm{from}\mathrm{the}\mathrm{formula}$$$${\mathrm{b}}_{\mathrm{k}}=2+7(\mathrm{k}-1),\mathrm{k}=1,2,...,102.\mathrm{The}\mathrm{problem}$$$$\mathrm{thus}\mathrm{consists}\mathrm{in}\mathrm{finding}\mathrm{all}\mathrm{numbers}\mathrm{n}$$$$\mathrm{and}\mathrm{k},1⩽\mathrm{n}⩽102,1⩽\mathrm{k}⩽102,\mathrm{for}\mathrm{which}$$$${\mathrm{a}}_{\mathrm{n}}={\mathrm{b}}_{\mathrm{k}},\mathrm{that}\mathrm{is},4\mathrm{n}+4=7\mathrm{k}$$$$\mathrm{From}\mathrm{the}\mathrm{equation}4(\mathrm{n}+1)=7\mathrm{k}\mathrm{it}\mathrm{is}$$$$\mathrm{evident}\mathrm{that}\mathrm{k}\mathrm{is}\mathrm{a}\mathrm{multiple}\mathrm{of}4,\mathrm{that}$$$$\mathrm{is},\mathrm{k}=4\mathrm{s}\mathrm{with}\mathrm{s}=\overline{1...25}(\mathrm{since}1⩽\mathrm{k}⩽102)$$$$.\mathrm{But}\mathrm{if}\mathrm{k}=4\mathrm{s}\mathrm{then}4(\mathrm{n}+1)=7.4\mathrm{s},\mathrm{or}$$$$\mathrm{n}=7\mathrm{s}-1.\mathrm{Since}1⩽\mathrm{n}⩽102,\mathrm{only}\mathrm{the}$$$$\mathrm{numbers}1,2,...,14\mathrm{are}\mathrm{permissible}$$$$\mathrm{values}\mathrm{of}\mathrm{s}.\mathrm{Thus},\mathrm{we}\mathrm{have}14\mathrm{numbers}$$$$\mathrm{that}\mathrm{are}\mathrm{common}\mathrm{to}\mathrm{both}\mathrm{A}.\mathrm{Ps}$$$$\mathrm{The}\mathrm{numbers}\mathrm{themselves}\mathrm{are}\mathrm{readily}$$$$\mathrm{found}\mathrm{either}\mathrm{from}\mathrm{the}\mathrm{formula}\mathrm{for}{\mathrm{a}}_{\mathrm{n}}$$$$\mathrm{when}\mathrm{n}=7\mathrm{s}-1,\mathrm{s}=\overline{1...14},\mathrm{or}\mathrm{from}\mathrm{the}$$$$\mathrm{formula}\mathrm{for}{\mathrm{b}}_{\mathrm{k}},\mathrm{for}\mathrm{k}=4\mathrm{s},\mathrm{s}=\overline{1...14}$$

Commented by PRITHWISH SEN 2 last updated on 19/Sep/20

$$\mathrm{great}\mathrm{sir}$$

Commented by 1549442205PVT last updated on 21/Sep/20

$$\mathrm{Thank}\mathrm{Sir}.\mathrm{You}\mathrm{are}\mathrm{welcome}.$$

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