Sequence and Series
Properties of Progressions
AIEEE 2005
Grade 11
Question:
Let $x = \sum_{n=0}^\infty a^n$, $y = \sum_{n=0}^\infty b^n$, $z = \sum_{n=0}^\infty c^n$ where $a, b, c \in (0, 1)$ and $a, b, c$ are in A.P. Then $x, y, z$ are in:
(1) A.P.
(2) G.P.
(3) H.P.
(4) Neither AP nor GP nor HP
Step-by-Step Solution
Key Concept: $x = 1/(1 - a)$, so $1/x = 1 - a$. Similarly $1/y = 1 - b, 1/z = 1 - c$. Since $a, b, c$ are in A.P., so are $1 - a, 1 - b, 1 - c$. What does that imply about $1/x, 1/y, 1/z$?
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (3)