Sequences And Series Ques 17
- If the first and the $(2 n-1)$ th term of an $\mathrm{AP}, \mathrm{GP}$ and $\mathrm{HP}$ are equal and their $n$ $th$ terms are $a, b$ and $c$ respectively, then
$(1988,2 \mathrm{M})$
(a) $a=b=c$
(b) $a \geq b \geq c$
(c) $a+c=b$
(d) $a c-b^2=0$
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Answer:
Correct Answer: 17.(a,b,d)
Solution: (a,b,d) Since, first and $(2 n-1)$ th terms are equal.
Let first term be $x$ and $(2 n-1)$ th term be $y$. whose middle term is $t_{n}$
Thus, in arithmetic progression, $t_n=\frac{x+y}{2}=a$
In geometric progression, $t_n=\sqrt{x y}=b$
In harmonic progression, $t_n=\frac{2 x y}{x+y}=c$
$\Rightarrow b^2=a c$ and $a>b>c \quad$ [ using $\mathrm{AM}>\mathrm{GM}>\mathrm{HM} $ ]
Here, equality holds (i.e. $a=b=c$ ) only if all terms are same.
Hence, options (a), (b) and (d) are correct.
BETA
