If the first term and the common difference of a GP are each...

If the first term and the common difference of a GP are each equal to $x+\sqrt{5-x^{2}}$ i, then the modulus of its nth term is
(a) $5^{n}$
(b) $5^{n-1}$
(c) $5^{\frac{n-1}{2}}$
(d) $5^{\frac{n}{2}}$

Key Concepts

Algebraic Simplification Involving Square Roots

Algebraic manipulation, especially simplifying expressions that involve square roots, is crucial in many mathematical problems. In the context of complex numbers, the combination of a variable term and a square root can often simplify to a constant when used in the modulus formula, demonstrating how complex expressions can reduce to simpler, often well-known, numerical forms.

Geometric Progression

A geometric progression is a sequence in which each term after the first is obtained by multiplying the previous term by a fixed non-zero number called the common ratio. The nth term of a geometric progression with first term a and common ratio r is given by a multiplied by r raised to the (n – 1) power. This formula is key when the same expression defines both the first term and the recurrence factor in the sequence.

Complex Number Modulus

The modulus (or absolute value) of a complex number, expressed as a + bi, is the distance of the complex number from the origin in the complex plane, calculated using the formula ?(a² + b²). This concept is essential when evaluating expressions involving complex numbers since it provides a real number measure of their size or magnitude.

Exponentiation of Absolute Values

When raising a complex number to a power, it is often useful to first determine its modulus. The modulus of a power of a complex number is the modulus of the complex number raised to that same power. This property is particularly important for sequences, such as geometric progressions, where both the term and the ratio may be complex.

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