JEE Main · 2024 · 01 Feb 2024 · Shift 1 · Sequences and Series

$3,a,b,c$ in A.P. with common difference $d$: $a=3+d,b=3+2d,c=3+3d$.
$3,a-1,b+1,c+9$ in G.P.: the terms are $3,2+d,4+2d,12+3d$. For a G.P., $(2+d{)}^2=3(4+2d)$:

For $d=-2$: the G.P. becomes $3,0,0,6$ — invalid, since a term of a G.P. cannot be zero. So $d=4$: $a=7,b=11,c=15$ (G.P. check: $3,6,12,24$, ratio 2 — valid).