Let α and β are the roots of the equation x² - x - 1 = 0. If pk = (α)ᵏ + (β)ᵏ, k ≥ 1 then which one

Q: Why can't I just use the quadratic formula to find $\alpha$ and $\beta$?

You absolutely could, but you would be walking into a trap! The roots are $\frac{1 \pm \sqrt{5}}{2}$. Raising these to the power of 5 involves binomial expansions that are prone to calculation errors. In JEE, we look for the 'elegant' path—using the properties of the equation itself to avoid brute force.

Q: How did you derive $p_k = p_{k-1} + p_{k-2}$ so quickly?

It comes from the characteristic equation. Since $\alpha^2 = \alpha + 1$, multiplying by $\alpha^{k-2}$ gives $\alpha^k = \alpha^{k-1} + \alpha^{k-2}$. Doing the same for $\beta$ and adding them yields the recurrence. It is a powerful shortcut that turns a complex power problem into simple addition.

Q: Is this sequence related to the Fibonacci sequence?

Spot on! You have discovered the Lucas numbers. The sequence $p_k$ follows the same recurrence as the Fibonacci sequence ($F_n = F_{n-1} + F_{n-2}$), just with different starting values. Recognizing these patterns is what separates a good student from a great one.

Question

Let $\alpha$ and $\beta$ are the roots of the equation $x^2 - x - 1 = 0$. If $p_k = (\alpha)^k + (\beta)^k, k \ge 1$ then which one of the following statements is not true?

Options:

(A) $(p_1 + p_2 + p_3 + p_4 + p_5) = 26$

(B) $p_5 = 11$

(C) $p_5 = p_2 \cdot p_3$

(D) $p_3 = p_5 - p_4$

Sigma Percentile · Accepted Answer

### The Hidden Elegance of Quadratic Sequences

Welcome, fellow traveler on the road to JEE success. Today, we are not just solving a quadratic equation; we are uncovering a hidden pattern that governs the behavior of numbers.

When you look at the equation $x^2 - x - 1 = 0$, what do you see? A simple quadratic? Or a gateway to a beautiful sequence? Let us peel back the layers together.

### Phase 1: The Foundation

Our journey begins with the roots $\alpha$ and $\beta$. Many students immediately reach for the quadratic formula, but I want you to pause.

In competitive exams, the most efficient path is rarely the most obvious one. We know from Vieta's formulas that for any quadratic $ax^2 + bx + c = 0$, the sum of roots is $-\frac{b}{a}$ and the product is $\frac{c}{a}$.

For our equation, this gives us:

$$\alpha + \beta = 1$$

$$\alpha\beta = -1$$

These two values are the keys to the kingdom. Hold onto them tightly.

### Phase 2: The Power of Recurrence

Now, we define $p_k = \alpha^k + \beta^k$. Calculating $p_5$ by raising the roots to the fifth power is a recipe for disaster.

Instead, let us use the 'Newton's Sums' approach. Since $\alpha$ is a root, it must satisfy the equation $\alpha^2 - \alpha - 1 = 0$, which rearranges to $\alpha^2 = \alpha + 1$.

If we multiply this entire equation by $\alpha^{k-2}$, we get:

$$\alpha^k = \alpha^{k-1} + \alpha^{k-2}$$

The same logic applies to $\beta$. When we add these two relations, we get the magic recurrence:

$$p_k = p_{k-1} + p_{k-2}$$

This is the heartbeat of our problem. Each term is simply the sum of the two preceding it.

### Phase 3: Building the Sequence

Let us calculate our terms step-by-step. We start with $p_1 = \alpha + \beta = 1$.

For $p_2$, we use the identity $\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta$. Substituting our known values, we get:

$$p_2 = (1)^2 - 2(-1) = 1 + 2 = 3$$

Now, the recurrence takes over:

- $p_3 = p_2 + p_1 = 3 + 1 = 4$

- $p_4 = p_3 + p_2 = 4 + 3 = 7$

- $p_5 = p_4 + p_3 = 7 + 4 = 11$

Look at that! We have generated the sequence $1, 3, 4, 7, 11$ without ever needing to touch a square root.

### Phase 4: The Final Verdict

Now, we test our options:

1. The sum $p_1 + p_2 + p_3 + p_4 + p_5 = 1 + 3 + 4 + 7 + 11 = 26$. This is true.

2. $p_5 = 11$. This is true.

3. $p_3 = p_5 - p_4 \Rightarrow 4 = 11 - 7$. This is true.

4. Finally, $p_5 = p_2 \cdot p_3 \Rightarrow 11 = 3 \cdot 4 = 12$. This is **clearly false!**

There you have it. By understanding the underlying structure of the roots rather than brute-forcing the calculation, we have dismantled the problem with surgical precision.

Keep this mindset—look for the pattern, trust the recurrence, and the answer will reveal itself.

Animated Solution for Mathematics - Quadratic Equations: Let $α$ and $β$ are the roots of the equation $x^{2} - x - 1 = 0$ . If $p_{k} = (α)^{k} + (β)^{k}, k \geq 1$ then which one of the following statements is not true?

Select Answer:

Visualized Solution

Introduction to $x^{2} - x - 1 = 0$

Sum and Product of Roots

Calculating $p_{1}$

Calculating $p_{2}$

Newton's Sums Recurrence

Calculating $p_{3}$

Calculating $p_{4}$

Calculating $p_{5}$

Checking Options 1 and 2

Checking Option 4

Checking Option 3

The Sigma Insight: Relation Between Roots and Coefficients

The Hidden Elegance of Quadratic Sequences

Phase 1

Phase 2

Phase 3

Phase 4

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