NIMCET 2022 — Mathematics PYQ

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Question

NIMCET 2022 — Mathematics PYQ

NIMCET | Mathematics | 2022

If $α$ and $β$ are roots of $x^{2} - x - 1 = 0$ , and $A_{n} = α^{n} + β^{n}$ , then the arithmetic mean of $A_{n - 1}$ and $A_{n}$ is

Choose the correct answer:

Explanation

1. Using the Property of Roots:

Since $\alpha$ and $\beta$ are the roots of the equation $x^2 - x - 1 = 0$ , they must satisfy it:

\alpha^2 - \alpha - 1 = 0 \implies \alpha^2 = \alpha + 1

\beta^2 - \beta - 1 = 0 \implies \beta^2 = \beta + 1

2. Deriving the Recurrence Relation:

We are given $A_n = \alpha^n + \beta^n$ . Let's look at the expression for $A_{n+1}$ :

A_{n+1} = \alpha^{n+1} + \beta^{n+1}

A_{n+1} = \alpha^{n-1}(\alpha^2) + \beta^{n-1}(\beta^2)

Substitute $\alpha^2 = \alpha + 1$ and $\beta^2 = \beta + 1$ into the equation:

A_{n+1} = \alpha^{n-1}(\alpha + 1) + \beta^{n-1}(\beta + 1)

A_{n+1} = (\alpha^n + \alpha^{n-1}) + (\beta^n + \beta^{n-1})

Rearrange the terms:

A_{n+1} = (\alpha^n + \beta^n) + (\alpha^{n-1} + \beta^{n-1})

A_{n+1} = A_n + A_{n-1}

3. Finding the Arithmetic Mean:

The Arithmetic Mean (A.M.) of $A_{n-1}$ and $A_n$ is defined as:

\text{A.M.} = \frac{A_{n-1} + A_n}{2}

From our recurrence relation $A_{n+1} = A_n + A_{n-1}$ , we can substitute the numerator:

\text{A.M.} = \frac{A_{n+1}}{2}

Correct Option:

B. $\frac{1}{2} A_{n+1}$

Choose the correct answer:

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