JEE MAIN - Mathematics (2024 - 27th January Morning Shift - No. 16)
The number of common terms in the progressions
$4,9,14,19, \ldots \ldots$, up to $25^{\text {th }}$ term and
$3,6,9,12, \ldots \ldots$, up to $37^{\text {th }}$ term is :
$4,9,14,19, \ldots \ldots$, up to $25^{\text {th }}$ term and
$3,6,9,12, \ldots \ldots$, up to $37^{\text {th }}$ term is :
9
8
5
7
Explanation
$$4,9,14,19, \ldots$$, up to $$25^{\text {th }}$$ term
$$\mathrm{T}_{25}=4+(25-1) 5=4+120=124$$
$$3,6,9,12, \ldots$$, up to $$37^{\text {th }}$$ term
$$\mathrm{T}_{37}=3+(37-1) 3=3+108=111$$
Common difference of $$\mathrm{I}^{\text {st }}$$ series $$\mathrm{d}_1=5$$
Common difference of $$\mathrm{II}^{\text {nd }}$$ series $$\mathrm{d}_2=3$$
First common term $$=9$$, and their common difference $$=15\left(\operatorname{LCM}\right.$$ of $$\mathrm{d}_1$$ and $$\left.\mathrm{d}_2\right)$$ then common terms are $$9,24,39,54,69,84,99$$
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