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If 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  are in A.P. and

\left | a \right |< 1,\left | b \right |< 1,\left | c \right |< 1\; then\; x,y,z\; are\; in

Option: 1

HP\;


Option: 2

AG-P


Option: 3

\; AP\;


Option: 4

\; GP


Answers (1)

best_answer

 x=\sum_{n=0}^{\infty}a^{n},\:y=\sum_{n=0}^{\infty}b^{n},\: z=\sum_{n=0}^{\infty}c^{n} 

and a, b, c re in A.P

\therefore x = 1 + a + a2 +.........\infty

\Rightarrow     x=\frac{1}{1-a}=1-a=\frac{1}{x}

y = 1 + b + b2 +........\infty

\Rightarrow    y=\frac{1}{1-b}=1-b=\frac{1}{y}

z = 1 + c + c2 +.........\infty 

\Rightarrow    z=\frac{1}{1-c}=1-c=\frac{1}{z}

Now, since a, b, c are in A.P

So -a, -b, -c are in A.P

So 1-a, 1-b, 1-c, are in A.P

So \frac{1}{x},\:\frac{1}{y}, \frac{1}{z} in A.P

So x, y, z are in H.P

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