If the first and the $ (2n - 1)th$ term of an $AP, GP$ and $HP$ are equal and their nth terms are $a, b$ and $c$ respectively, then
- $a = b = c$
- $a \ge b \ge c$
- $a + c = b$
- $ac - b^2 =0$
The Correct Option is D
Solution and Explanation
Let first term be x and (2n - 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, $t_n =\sqrt {xy} = b$
In harmonic progression, $t_n =\frac{2xy}{x+y} =c $
$ \Rightarrow \, \, \, b^2 = ac $ and $a > b >c \, \, \, \, \, \, [using\, AM > GM > HM] $
Here, equality holds (i.e. a = b =c) only if all terms are
same. Hence, options (a), (b) and (d) are correct
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Concepts Used:
Sequence and Series
Sequence: Sequence and Series is one of the most important concepts in Arithmetic. A sequence refers to the collection of elements that can be repeated in any sort.
Eg: a1,a2,a3, a4…….
Series: A series can be referred to as the sum of all the elements available in the sequence. One of the most common examples of a sequence and series would be Arithmetic Progression.
Eg: If a1,a2,a3, a4……. etc is considered to be a sequence, then the sum of terms in the sequence a1+a2+a3+ a4……. are considered to be a series.
Types of Sequence and Series:
Arithmetic Sequences
A sequence in which every term is created by adding or subtracting a definite number to the preceding number is an arithmetic sequence.
Geometric Sequences
A sequence in which every term is obtained by multiplying or dividing a definite number with the preceding number is known as a geometric sequence.
Harmonic Sequences
A series of numbers is said to be in harmonic sequence if the reciprocals of all the elements of the sequence form an arithmetic sequence.
Fibonacci Numbers
Fibonacci numbers form an interesting sequence of numbers in which each element is obtained by adding two preceding elements and the sequence starts with 0 and 1. Sequence is defined as, F0 = 0 and F1 = 1 and Fn = Fn-1 + Fn-2