Q. 94 聽Let akbe a sequence. Prove The... [FREE SOLUTION]

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Chapter 7: Q. 94 (page 593)

Let $\{a_{k}\}$ be a sequence. Prove Theorem 7.6 (a) along with the following variations:
(a) Show that when role="math" localid="1649277359535" $a_{k - 1} - a_{k}$ 鈮� 0 for every k 鈮� 1, the sequence is increasing.
(b) Show that when $a_{k - 1} - a_{k}$ > 0 for every k 鈮� 1, the sequence is strictly increasing.
(c) Show that when role="math" localid="1649277346412" $a_{k - 1} - a_{k}$ 鈮� 0 for every k 鈮� 1, the sequence is decreasing.
(d) Show that when $a_{k - 1} - a_{k}$ < 0 for every k 鈮� 1, the sequence is strictly decreasing.

Short Answer

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Proved

Step by step solution

Step 1. Given

Consider the sequence $\{a_{k}\}$

Part (a) Step 2. Explanation

$Since, a_{k + 1} - a_{k} 鈮� 0, now all the terms of the sequence are positive, therefore a_{k} is also positive . Now, adding the inequality a_{k + 1} - a_{k} 鈮� 0 with a_{k} . The sign of the inequality will not change as a_{k} is positive . Now, a_{k + 1} - a_{k} + a_{k} 鈮� 0 + a_{k} a_{k + 1} 鈮� a_{k} for k 鈮� 1 Hence, be definition the sequence is i n creasing sequence .$

Part(b) Step 3. Explanation

$Since, a_{k + 1} - a_{k} > 0, now all the terms of the sequence are positive, therefore a_{k} is also positive . Now, adding the inequality a_{k + 1} - a_{k} > 0 with a_{k} . The sign of the inequality will not change as a_{k} is positive . Now, a_{k + 1} - a_{k} + a_{k} > 0 + a_{k} a_{k + 1} > a_{k} for k 鈮� 1 Hence, be definition the sequence is strictly i n creasing sequence .$

Part(c) Step 4. Explanation

Part(d) Step 5. Explanation

$Since, a_{k + 1} - a_{k} < 0, now all the terms of the sequence are positive, therefore a_{k} is also positive . Now, adding the inequality a_{k + 1} - a_{k} < 0 with a_{k} . The sign of the inequality will not change as a_{k} is positive . Now, a_{k + 1} - a_{k} + a_{k} < 0 + a_{k} a_{k + 1} < a_{k} for k 鈮� 1 Hence, be definition the sequence is s t r i c t l y d e creasing sequence .$

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Short Answer

Step by step solution

Step 1. Given

Part (a) Step 2. Explanation

Part(b) Step 3. Explanation

Part(c) Step 4. Explanation

Part(d) Step 5. Explanation

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