If and ⟨bn>. be two sequences given by an= x^2-+y^2 and bn=x^2^-4-y^2^-4 ∀n ∈N, then the va

step 1 Here we have bn is equal to x to the power 2 to the power minus n minus 1 whole square minus y t

If <an> and ⟨bn>. be two sequences given by an= x^2-+y^2 and...

If $<a_{n}>$ and $\left\langle b_{n}>\right.$ be two sequences given by $a_{n}=$ $x^{2-}+y^{2}$ and $b_{n}=x^{2^{-4}}-y^{2^{-4}} \forall n \in N$, then the value of $a_{1} a_{2} a_{3} \ldots a_{n}$ is (A) $\frac{x+y}{b_{n}}$ (B) $\frac{x-y}{b_{n}}$ (C) $\frac{x^{2}+y^{2}}{b_{n}}$ (D) $\frac{x^{2}-y^{2}}{b_{n}}$

If $<a_{n}>$ and $\left\langle b_{n}>\right.$ be two sequences given by $a_{n}=$ $x^{2-}+y^{2}$ and $b_{n}=x^{2^{-4}}-y^{2^{-4}} \forall n \in N$, then the
value of $a_{1} a_{2} a_{3} \ldots a_{n}$ is
(A) $\frac{x+y}{b_{n}}$
(B) $\frac{x-y}{b_{n}}$
(C) $\frac{x^{2}+y^{2}}{b_{n}}$
(D) $\frac{x^{2}-y^{2}}{b_{n}}$

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Key Concepts

Telescoping Products

Telescoping products are expressions in which each term in a product cancels with part of another term, simplifying the overall computation. In many cases, when a product is written out, a pattern emerges that allows intermediate factors to cancel, leaving only a few terms from the beginning and the end. This concept is widely used to simplify complicated sequences and series in algebra and calculus.

Difference of Squares Identity

The difference of squares identity states that a² - b² = (a + b)(a - b). This factorization is fundamental in algebra because it helps simplify expressions that, on the surface, appear complex by breaking them down into simpler multiplicative components. The identity is often a key tool in rearranging and telescoping products, particularly when terms can be manipulated to reveal a cancelling structure.

Exponential Sequences

Exponential sequences involve terms where variables are raised to powers that are functions of the sequence index. Understanding the properties of exponents, such as how to manage and simplify expressions with exponential growth or decay, is crucial for handling products and sums involving such sequences. This manipulation often leads to factorization or telescoping patterns that significantly simplify the evaluation of the overall product.

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