Question

If the first and $\left( 2n-1 \right){{~}^{th}}$ terms of an A.P, G.P and H.P are equal and their ${{n}^{th}}$ terms are respectively a, b and c then always
A) $a\text{ }=\text{ }b\text{ }=\text{ }c$
B) $a\ge ~b~\ge c$
C) $a+\text{ }c\text{ }=\text{ }b$
D) $ac-\text{ }{{b}^{2}}~=\text{ }0$

Answer 1

Find the value of ${{n}^{th}}$term in terms of first and $\left( 2n-1 \right){{~}^{th}}$term for AP, GP and HP series using the formulas of the ${{n}^{th}}$ term by letting the common difference: d and common ratio: r, for the respective series. Then try to relate them to get the answer to this question.

Formula Used:
For A.P series we know that,
${{n}^{th}}$= first term + (n-1) common difference

Now, for GP series we know that the ${{n}^{th}}$term
$=\text{ }first\text{ }term~{{\left( common\text{ }ratio \right)}^{n-1}}$

Complete step by step solution:
It is given that the first and $\left( 2n-1 \right){{~}^{th}}$terms of AP, GP and HP are equal.
Let the first term be x and $\left( 2n-1 \right){{~}^{th}}$term be y
For A.P series we know that,
${{n}^{th}}$= first term + (n-1) common difference
The first term is x and $\left( 2n-1 \right){{~}^{th}}$term is y
And let the common difference = d
$\Rightarrow$y= x + (2n-2) d
The ${{n}^{th}}$= a$=~x+\text{ }\left( n-1 \right)\text{ }d$ (1)
{it is given that ${{n}^{th}}$term of AP series =a }
Now,
The sum of first term and $\left( 2n-1 \right){{~}^{th}}$term
= x+ y
= x+ x + (2n-2) d
= 2x + (2n-2) d
= 2[x+(n-1)d]
From (1)
= 2[a]
$\Rightarrow$x+ y = 2[a]
$\Rightarrow$ (x+ y)/2 = a
Or,
$\Rightarrow$ a = (x+ y)/2 (2)

Now, for GP series we know that the ${{n}^{th}}$ term
= first term $\times$ (common ratio)$^{n-1}$
Let the common ratio = r
$\Rightarrow$ ${{n}^{th}}$ term$\begin{aligned} & =\text{ }x~{{r}^{n-1}}~=\text{ }b \\\ & ~ \\\ \end{aligned}$ (3)
{it is given in that the ${{n}^{th}}$ term of GP series =b}
Now,
(2n-1) $^{th}$ term = $x~{{r}^{2n-2}}$
$\Rightarrow$first term $\times$ (2n-1) $^{th}$ term = $x\times ~x\times ~{{r}^{2n-2}}$= ${{x}^{2}}\times {{r}^{2n-2}}$
= ${{(x\times {{r}^{n-1}})}^{2}}$
From (3)
$\Rightarrow x~\times y\text{ }=\text{ }b{{~}^{2}}$
Taking square root on both sides
We get,
$\sqrt{xy}=b$ (4)
Now, for HP series,
A sequence of numbers is called a harmonic progression if the reciprocal of the terms are in AP. In simple terms a, b, c, d, e, f are in HP if 1/a, 1/b, 1/c, 1/d, 1/e, 1/f are in AP.
For two terms that is x and y
x= first term,
y= $\left( 2n-1 \right){{~}^{th}}$term
Harmonic Mean, c = (2 x y) / (x + y)
(5)
From (2), (4) and (5) the conditions satisfied are

& a\text{ }\ge ~b\text{ }\ge \text{ }c \\\ & a\text{ }b\text{ }=\text{ }{{c}^{2}} \\\ & \Rightarrow ab-{{c}^{2}}~=\text{ }0 \\\ \end{aligned}$$ Hence, **B) and C) are correct.** **Additional Information:** Progressions (or Sequences and Series) are numbers arranged in a particular order such that they form a predictable order. By predictable order, we mean that given some numbers, we can find the next numbers in the series. Arithmetic Progression (AP): A sequence of numbers is called an arithmetic progression if the difference between any two consecutive terms is always the same. If ‘a’ is the first term and d is the common difference, nth term of an AP = a + (n-1) d Geometric Progression (GP): A sequence of numbers is called a geometric progression if the ratio of any two consecutive terms is always the same. If ‘a’ is the first term and ‘r’ is the common ratio, nth term of a GP= a rn-1 Sum of ‘n’ terms of a GP(r<1) $$=\text{ }\left[ a\text{ }\left( 1\text{ }\text{ }{{r}^{n}} \right)\left] \text{ }/\text{ } \right[1\text{ }\text{ }r \right]$$ Sum of ‘n’ terms of a GP (r > 1)$$=\text{ }\left[ a\text{ }\left( {{r}^{n}}~\text{ }1 \right)\left] \text{ }/\text{ } \right[r\text{ }\text{ }1 \right]$$ Sum of infinite terms of a GP (r < 1) = (a) / (1 – r) Harmonic Progression (HP): A sequence of numbers is called a harmonic progression if the reciprocal of the terms are in AP. In simple terms, a, b, c, d, e, f are in HP if 1/a, 1/b, 1/c, 1/d, 1/e, 1/f are in AP. For two terms ‘a’ and ‘b’, Harmonic Mean = (2 a b) / (a + b) For two numbers, if A, G and H are respectively the arithmetic, geometric and harmonic means, then A ≥ G ≥ H A H = G2, i.e., A, G, H are in GP **Note:** The knowledge of progressions series that is AP, GP and HP series that is formula for $${{n}^{th}}$$ term of these series which are If ‘a’ is the first term and d is the common difference, $${{n}^{th}}$$term of an AP = a + (n-1) d If ‘a’ is the first term and ‘r’ is the common ratio, $${{n}^{th}}$$term of a GP = a rn-1 is important for students to answer this question.