Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

For an arbitrary positive integer $n \geq 3$ , find all solutions $x_{1}, x_{2}, x_{3}, . ....., x_{n}$ of the simultaneous equations $x_{2} = \frac{1}{2} (x_{1} + x_{3}), x_{3} = \frac{1}{2} (x_{2} + x_{4}), . ...., x_{n - 1} = \frac{1}{2} (x_{n - 2} + x_{n})$ . Note that we are asked to solve the simultaneous equations $x_{k} = \frac{1}{2} (x_{k - 1} + x_{k + 1})$ , for $k = 2, 3, . ...., n - 1$ .

$x_{n} = 3 x_{n + 2} - 2 x_{n + 3}$ is the expression using which the solutions $x_{1}, x_{2}, x_{3}, ......, x_{n}$ of the simultaneous equations can be obtained.

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Consider the given equations.

The expression for $x_{2}$ is,

$x_{2} = \frac{1}{2} (x_{1} + x_{3})$

Substitute the above equation in $x_{3}$

$\begin{array}{l} x_{3} = \frac{1}{2} (x_{2} + x_{4}) \\ \Rightarrow x_{3} = \frac{1}{2} ((\frac{1}{2} (x_{1} + x_{3})) + x_{4}) \\ \Rightarrow x_{3} = \frac{1}{4} x_{1} + \frac{1}{2} x_{3} + \frac{1}{2} x_{4} \\ ∴ x_{1} = 3 x_{3} - 2 x_{4} \end{array}$

Step 2: Compute the expressions for remaining variables.

Similarly,

$x_{2} = 3 x_{4} - 2 x_{5}$

Continuing the same, the expression for the $n^{t h}$ term will be,

$x_{n} = 3 x_{n + 2} - 2 x_{n + 3}$

Step 3: Compute the expressions for remaining variables.

Similarly,

$x_{2} = 3 x_{4} - 2 x_{5}$

Continuing the same, the expression for the $n^{t h}$ term will be,

x_{n} = 3 x_{n + 2} - 2 x_{n + 3}

The solutions $x_{1}, x_{2}, x_{3}, ......, x_{n}$ of the simultaneous equations can be found using the expression, $x_{n} = 3 x_{n + 2} - 2 x_{n + 3}$ .

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Linear Algebra With Applications

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Short Answer

Step by Step Solution

Step 1: Consider the given equations.

Step 2: Compute the expressions for remaining variables.

Step 3: Compute the expressions for remaining variables.

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Linear Algebra With Applications

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Short Answer

For an arbitrary positive integern≥3 , find all solutions x1,x2,x3,......,xnof the simultaneous equations x2=12(x1+x3),x3=12(x2+x4),.....,xn−1=12(xn−2+xn). Note that we are asked to solve the simultaneous equations xk=12(xk−1+xk+1), for k=2,3,.....,n−1 .

Step by Step Solution

Step 1: Consider the given equations.

Step 2: Compute the expressions for remaining variables.

Step 3: Compute the expressions for remaining variables.

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