Solve it by using induction method: **Exercise: Mathematical Induction****Problem Statement:**Prove that for every integer $ n \geq 0 $, $ 9 \mid (4^{3n} + 8) $.**Solution Approach:**To prove this statement, you may consider using the method of mathematical induction. The process involves the following steps:1. **Base Case:** Verify the statement for the initial value $ n = 0 $.2. **Inductive Step:** Assume the statement is true for $ n = k $, i.e., $ 9 \mid (4^{3k} + 8) $. Then, prove it for $ n = k + 1 $.**Explanation:**The notation $ 9 \mid (4^{3n} + 8) $ means that the expression $ 4^{3n} + 8 $ is divisible by 9 for every integer $ n $ greater than or equal to 0.You will find that both steps of induction confirm the divisibility condition, solidifying the statement for all non-negative integers $ n $.

To prove: For every integer n≥0, 9|43n+8. Proof by induction method.

Answered: 3n Prove that for every integer n > 0,…

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3n Prove that for every integer n > 0, 9|(4" + 8).

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Matrices

‘Matrices’ is the plural form of the word matrix, and it is basically a spreadsheet in the form of a box. In mathematics, various functions can be carried out with matrices. Generally, a matrix comes in the shape of a square or rectangle. The elements are distributed horizontally in the form of rows, and vertically in the form of columns.

Question

Solve it by using induction method:

**Exercise: Mathematical Induction**

**Problem Statement:**

Prove that for every integer $ n \geq 0 $, $ 9 \mid (4^{3n} + 8) $.

**Solution Approach:**

To prove this statement, you may consider using the method of mathematical induction. The process involves the following steps:

1. **Base Case:** Verify the statement for the initial value $ n = 0 $.
2. **Inductive Step:** Assume the statement is true for $ n = k $, i.e., $ 9 \mid (4^{3k} + 8) $. Then, prove it for $ n = k + 1 $.

**Explanation:**

The notation $ 9 \mid (4^{3n} + 8) $ means that the expression $ 4^{3n} + 8 $ is divisible by 9 for every integer $ n $ greater than or equal to 0.

You will find that both steps of induction confirm the divisibility condition, solidifying the statement for all non-negative integers $ n $.

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