Read the data carefully and answer the following question:

Given the sequence: $$20, a, b, c, d, 65$$

The terms satisfy the following relations:

- $$a - 20 = X^2 + Y$$
- $$b - a = (X + 1)^2 + Y \ (b a)$$
- $$c - b = (X + 2)^2 + Y \ (c b)$$
- $$d - c = (X + 3)^2 + Y \ (d c)$$

Where:

- $$X$$ is the HCF of two prime numbers $$M$$ and $$N$$
- $$Y$$ is the smallest root of the quadratic equation $$z^2 - z - 6 = 0$$

Find the value of $$d$$.

Question

Read the data carefully and answer the following question:

Given the sequence: $$20, a, b, c, d, 65$$

The terms satisfy the following relations:

- $$a - 20 = X^2 + Y$$
- $$b - a = (X + 1)^2 + Y \ (b > a)$$
- $$c - b = (X + 2)^2 + Y \ (c > b)$$
- $$d - c = (X + 3)^2 + Y \ (d > c)$$

Where:

- $$X$$ is the HCF of two prime numbers $$M$$ and $$N$$
- $$Y$$ is the smallest root of the quadratic equation $$z^2 - z - 6 = 0$$

Find the value of $$d$$.

Accepted Answer

**Solution:**

1. First, find the value of $$Y$$ by solving the quadratic equation:
   $$z^2 - z - 6 = 0$$

Using the quadratic formula:
   $$z = \frac{1 \pm \sqrt{1 + 24}}{2} = \frac{1 \pm 5}{2}$$

So, the roots are:
   - $$z = \frac{1 + 5}{2} = 3$$
   - $$z = \frac{1 - 5}{2} = -2$$

The smallest root is $$Y = -2$$.

2. Since $$X$$ is the HCF of two prime numbers $$M$$ and $$N$$, and both are prime, their HCF can only be 1 if they are different primes (since prime numbers have no common factors other than 1).

Therefore, $$X = 1$$.

3. Now, calculate each term difference:

- $$a - 20 = X^2 + Y = 1^2 + (-2) = 1 - 2 = -1$$
     $$\Rightarrow a = 20 - 1 = 19$$

- $$b - a = (X + 1)^2 + Y = (1 + 1)^2 - 2 = 2^2 - 2 = 4 - 2 = 2$$
     $$\Rightarrow b = a + 2 = 19 + 2 = 21$$

- $$c - b = (X + 2)^2 + Y = (1 + 2)^2 - 2 = 3^2 - 2 = 9 - 2 = 7$$
     $$\Rightarrow c = b + 7 = 21 + 7 = 28$$

- $$d - c = (X + 3)^2 + Y = (1 + 3)^2 - 2 = 4^2 - 2 = 16 - 2 = 14$$
     $$\Rightarrow d = c + 14 = 28 + 14 = 42$$

4. However, the last term given is 65, so let's verify the sequence:

- The sequence is: $$20, 19, 21, 28, 42, 65$$

Check if $$65 - d$$ fits the pattern:

The problem does not specify the difference between $$d$$ and 65, so we consider $$d$$ as the fifth term.

5. The question asks for the value of $$d$$, which we calculated as $$42$$.

**Answer:** $42$

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