AMC 10A 2025 (Problem 11)The sequence 1,x,y,z1,x,y,z1,x,y,z is arithmetic. The sequence 1,p,q,z1,p,q,z1,p,q,z is geometric. Both sequences are strictly increasing and contain only integers, and zzz is as small as possible. What is the value of x+y+z+p+qx+y+z+p+qx+y+z+p+q?(A) 66\text{(A)}\;66(A)66(B) 91\text{(B)}\;91(B)91(C) 103\text{(C)}\;103(C)103(D) 132\text{(D)}\;132(D)132(E) 149\text{(E)}\;149(E)149Related TopicsCoreToolkit 25 — Arithmetic Sequence and SeriesToolkit 4 — Geometric Sequence and SeriesMinorToolkit 47 — Modular Arithmetic: Definition and PropertiesToolkit 75 — Try Small ExamplesHints (7)Hint 11,x,y,z:1,1+d,1+2d,1+3d(d∈Z+)1,x,y,z: \quad 1,1+d,1+2d,1+3d \quad (d\in\mathbb{Z}^+)1,x,y,z:1,1+d,1+2d,1+3d(d∈Z+)Hint 21,p,q,z:1,r,r2,r3(r∈Z+, r>1)1,p,q,z: \quad 1,r,r^2,r^3 \quad (r\in\mathbb{Z}^+,\ r>1)1,p,q,z:1,r,r2,r3(r∈Z+, r>1)Hint 31+3d=z=r31+3d=z=r^31+3d=z=r3Hint 4r3≡1(mod3)⟹r≡1(mod3)r^3\equiv1\pmod{3}\Longrightarrow r\equiv1\pmod{3}r3≡1(mod3)⟹r≡1(mod3)Hint 5So, the first rrr that we should check is r=4r=4r=4.Hint 61+3d=z=r3=64⟹d=211+3d=z=r^3=64\Longrightarrow d=211+3d=z=r3=64⟹d=21Hint 71,x,y,z:1,22,43,641,x,y,z:\quad 1,22,43,641,x,y,z:1,22,43,641,p,q,z:1,4,16,641,p,q,z:\quad 1,4,16,641,p,q,z:1,4,16,64x+y+z+p+q=22+43+64+4+16=149x+y+z+p+q=22+43+64+4+16=149x+y+z+p+q=22+43+64+4+16=149Final Answer(E) 149149149Related Problems (4)AMC 10B 2022 (Problem 15)AIME II 2026 (Problem 9)MATHCOUNTS 2026 State Target Round (Problem 5)AMC 10A/12A 2025 (Problem 13/5)