Which of the following statement (s) is (are) correct ?

Options

A. Sum of the reciprocal of all the n harmonic means inserted between a and b is equal to n times the harmonic mean between two given numbers a and b.
B. Sum of the cubes of first n natural number is equal to square of the sum of the first a natural numbers.
C. If `a , A_(1), A_(2), A_(3), ….., A_(2n), b` are in A.P. then `sum _( I =1) ^(2n) A_(l) =n (a+b).`
D. If the first term of the geometric progression `g _(1), g _(2), g _(3), ……, oo` is unity, then the value of the common ratio of the progression such that `(4g _(2)+5g _(3))` is minimum equals `2/5.`

Answer

To solve the question regarding the correctness of the given statements, we will analyze each statement step by step.

### Step 1: Analyze the first statement
The first statement claims that the sum of the reciprocals of all n harmonic means inserted between two numbers \( a \) and \( b \) is equal to \( n \) times the harmonic mean of \( a \) and \( b \).

1. **Understanding Harmonic Means**: If \( h_1, h_2, \ldots, h_n \) are the harmonic means between \( a \) and \( b \), then they are in harmonic progression (HP). The relationship between the harmonic means and their reciprocals is that the reciprocals will be in arithmetic progression (AP).

2. **Reciprocal Sum**: The sum of the reciprocals of the harmonic means plus the reciprocals of \( a \) and \( b \) can be expressed as:
\[ S = \frac{1}{a} + \frac{1}{h_1} + \frac{1}{h_2} + \ldots + \frac{1}{h_n} + \frac{1}{b} \]
Since \( \frac{1}{a}, \frac{1}{h_1}, \ldots, \frac{1}{h_n}, \frac{1}{b} \) are in AP, we can use the formula for the sum of an AP:
\[ S = \frac{(n + 2)}{2} \left( \frac{1}{a} + \frac{1}{b} \right) \]

3. **Harmonic Mean Calculation**: The harmonic mean \( H \) of \( a \) and \( b \) is given by:
\[ H = \frac{2ab}{a + b} \]
Therefore, \( n \times H = n \times \frac{2ab}{a + b} \).

4. **Comparison**: The statement claims:
\[ \frac{(n + 2)}{2} \left( \frac{1}{a} + \frac{1}{b} \right) = n \times \frac{2ab}{a + b} \]
This leads to a contradiction, as we find that the left side does not equal the right side for all \( a \) and \( b \). Hence, the first statement is **incorrect**.

### Step 2: Analyze the second statement
The second statement claims that the sum of the cubes of the first \( n \) natural numbers is equal to the square of the sum of the first \( n \) natural numbers.

1. **Sum of Cubes**: The formula for the sum of cubes of the first \( n \) natural numbers is:
\[ S_1 = 1^3 + 2^3 + 3^3 + \ldots + n^3 = \left( \frac{n(n + 1)}{2} \right)^2 \]

2. **Square of the Sum**: The sum of the first \( n \) natural numbers is:
\[ S_2 = 1 + 2 + 3 + \ldots + n = \frac{n(n + 1)}{2} \]
Therefore, the square of this sum is:
\[ S_2^2 = \left( \frac{n(n + 1)}{2} \right)^2 \]

3. **Conclusion**: Since \( S_1 = S_2^2 \), the second statement is **correct**.

### Step 3: Analyze the third statement
The third statement claims that \( a_1, a_2, \ldots, a_n, b \) are in arithmetic progression (AP).

1. **AP Definition**: For \( a_1, a_2, \ldots, a_n \) to be in AP with \( b \), the condition is that the common difference remains constant.

2. **Sum of Terms**: The sum of the terms can be expressed as:
\[ S = a_1 + a_2 + \ldots + a_n + b \]
If we denote the first term as \( a \) and the last term as \( b \), the sum can be expressed as:
\[ S = \frac{(n + 1)}{2} (a + b) \]

3. **Conclusion**: The statement holds true as the terms can be arranged to maintain a constant difference. Thus, the third statement is **correct**.

### Step 4: Analyze the fourth statement
The fourth statement claims that if the first term of a geometric progression (GP) is unity, then the value of the common ratio such that \( 4G_2 + 5G_3 \) is minimized is \( \frac{2}{5} \).

1. **Geometric Progression Terms**: Let the first term \( G_1 = 1 \), \( G_2 = r \), \( G_3 = r^2 \).

2. **Expression**: The expression to minimize is:
\[ f(r) = 4r + 5r^2 \]

3. **Finding Minimum**: To find the minimum, we differentiate:
\[ f'(r) = 4 + 10r \]
Setting \( f'(r) = 0 \) gives:
\[ 10r + 4 = 0 \implies r = -\frac{2}{5} \]
However, since \( r \) must be positive in a GP, we check the second derivative to confirm that this is a minimum point.

4. **Conclusion**: The statement claims \( r = \frac{2}{5} \) is incorrect as the critical point found is negative. Thus, the fourth statement is **incorrect**.

### Final Conclusion
- **First Statement**: Incorrect
- **Second Statement**: Correct
- **Third Statement**: Correct
- **Fourth Statement**: Incorrect