Problem B.1Consider the following sequence of successive numbers of the 2*-th power:1, 22", 32*, 42*, 52",Show that the difference between the numbers in this sequence is odd for all k E N.

Answered: lem B.1 ler the following

Algebra for College Students

10th Edition

ISBN:9781285195780

Author:Jerome E. Kaufmann, Karen L. Schwitters

Publisher:Jerome E. Kaufmann, Karen L. Schwitters

Chapter14: Sequences And Mathematical Induction

Section14.4: Mathematical Induction

Problem 11PS

See similar textbooks

Related questions

Concept explainers

Question

Problem B.1
Consider the following sequence of successive numbers of the 2*-th power:
1, 22", 32*, 42*, 52",
Show that the difference between the numbers in this sequence is odd for all k E N.

Expert Solution

Step 1

The sequence of successive numbers is given,

$1, 2^{2^{k}}, 3^{2^{k}}, 4^{2^{k}}, 5^{2^{k}} . . . . . . . . . .$

which is expressed as ,

$1, 2^{2^{k}}, 3^{2^{k}}, 4^{2^{k}}, 5^{2^{k}} . . . . . . . . . . . . . n^{2^{k}}, {(n + 1)}^{2^{k}} .$

For $k = 1,$

Difference between two consecutive numbers,

$\begin{array}{rcl} {(n + 1)}^{2^{k}} - n^{2^{k}} & = & {(n + 1)}^{2} - n^{2} \\ = & n^{2} + 2 n + 1 - n^{2} \\ = & 2 n + 1 . \end{array}$

Therefore, $2 \begin{array}{rcl} n \end{array} \begin{array}{rcl} + \end{array} \begin{array}{rcl} 1 \end{array}$ is always an odd number.

Step 2

Suppose the result is true for $k = m,$

Therefore, ${(n + 1)}^{2^{m}} - n^{2^{m}}$ is always odd.

Let, ${(n + 1)}^{2^{m}} - n^{2^{m}} = 2 p + 1 .$ , $p \in Z .$

${(n + 1)}^{2^{m}} = n^{2^{m}} + 2 p + 1 . . . . . . . . . . . . . . . . . . . . . (1)$

Now, for $k = m + 1 .$

Difference ,

$\begin{array}{rcl} {(n + 1)}^{2^{m + 1}} - n^{2^{m + 1}} & = & {(n + 1)}^{2^{m \cdot 2}} - n^{2^{m + 1}} . \\ = & {\{{(n + 1)}^{2^{m}}\}}^{2} - n^{2^{m + 1}} . \\ = & {\{n^{2^{m}} + 2 p + 1\}}^{2} - n^{2^{m + 1}} . \\ = & n^{2^{m} \cdot 2} + 2 \{n^{2^{m}} \cdot (2 p + 1)\} + \{4 p^{2} + 4 p + 1\} - n^{2^{m} \cdot 2} \\ = & 2 \cdot \{n^{2^{m}} \cdot (2 p + 1) + 2 p^{2} + 2 p\} + 1 \\ = & 2 S + 1 . \end{array}$

Step by step

Solved in 3 steps

Knowledge Booster

$Linear Equations$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

Recommended textbooks for you

Algebra for College Students

Algebra

ISBN:

9781285195780

Author:

Jerome E. Kaufmann, Karen L. Schwitters

Publisher:

Cengage Learning