About Prove each of the following statements using mathematical induction. (a) Prove that for any positive integer n, 4 evenly divides 32n-1

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About Prove each of the following statements using mathematical induction. (a) Prove that for any positive integer n, 4 evenly divides 32n-1. (b) Prove that for any positive integer n, 6 evenly divides 7n – 1. (c) Prove that for any positive integer n, 4 evenly divides 11n – 7n. (d) Prove that for any positive integer n, 7 evenly divides 9n – 2n.

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Thành Đạt 5 months 2021-08-29T20:51:17+00:00 1 Answers 0 views 0

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    2021-08-29T20:52:32+00:00

    Answer:

    (a) 3^{2{k+1}} - 1 = 4(9k + 2)

    (b) 7^{k+1} - 1  = 6(7k + 1)

    (c) 11(k + 1) - 7(k + 1) = 4(k + 1)

    (d) 9(k+1) - 2(k+1) = 7(k+1)

    Step-by-step explanation:

    Solving (a): For integer n, 4 divides 3^{2n} - 1

    The proof is as follows:

    For n = k, we have:

    \frac{3^{2k}- 1}{4} = k

    Multiply through by 4

    3^{2k} - 1 = 4k

    9^k - 1 = 4k

    Next, prove the statement is true for n = k+1

    3^{2{k+1}} - 1

    3^{2{k+1}} - 1 = 9^{{k+1}} - 1

    3^{2{k+1}} - 1 = 9^{k} * 9^1 - 1

    3^{2{k+1}} - 1 = 9^k * 9 - 1

    Express -1 as – 9 + 8

    3^{2{k+1}} - 1 = 9^k * 9 - 9 + 8

    Factorize:

    3^{2{k+1}} - 1 = 9(9^k - 1) + 8

    Recall that: 9^k - 1 = 4k

    So, we have:

    3^{2{k+1}} - 1 = 9*4k + 8

    Factorize

    3^{2{k+1}} - 1 = 4(9k + 2)

    Since the above mathematical statement is true, then the given statement has been proved

    Solving (b): For integer n, 6 divides 7^n - 1

    The proof is as follows:

    For n = k, we have:

    \frac{7^k - 1}{6} = k

    Multiply through by 6

    7^k - 1 = 6k

    Next, prove the statement is true for n = k+1

    7^{k+1} - 1

    7^{k+1} - 1  = 7^k * 7^1 - 1

    7^{k+1} - 1  = 7^k * 7 - 1

    Express -1 as – 7 + 6

    7^{k+1} - 1  = 7^k * 7 - 7 + 6

    Factorize:

    7^{k+1} - 1  = 7(7^k  - 1) + 6

    Recall that: 7^k - 1 = 6k

    So, we have:

    7^{k+1} - 1  = 7(6k) + 6

    Factorize

    7^{k+1} - 1  = 6(7k + 1)

    Since the above mathematical statement is true, then the given statement has been proved

    Solving (c): For integer n, 4 divides 11n - 7n

    The proof is as follows:

    For n = k, we have:

    \frac{11k - 7k}{4} = k

    Multiply through by 4

    11k - 7k = 4k

    Next, prove the statement is true for n = k+1

    11(k + 1) - 7(k + 1)

    11(k + 1) - 7(k + 1) = 11k + 11 - 7k - 7

    Collect Like Terms

    11(k + 1) - 7(k + 1) = 11k - 7k + 11 - 7

    11(k + 1) - 7(k + 1) = 11k - 7k + 4

    Recall that: 11k - 7k = 4k

    So, we have:

    11(k + 1) - 7(k + 1) = 4k + 4

    Factorize

    11(k + 1) - 7(k + 1) = 4(k + 1)

    Since the above mathematical statement is true, then the given statement has been proved

    Solving (d): For integer n, 7 divides 9n - 2n

    The proof is as follows:

    For n = k, we have:

    \frac{9k - 2k}{7} = k

    Multiply through by 7

    9k - 2k = 7k

    Next, prove the statement is true for n = k+1

    9(k+1) - 2(k+1)

    9(k+1) - 2(k+1) = 9k+9 - 2k-2

    Collect Like Terms

    9(k+1) - 2(k+1) = 9k- 2k+9 -2

    9(k+1) - 2(k+1) = 9k- 2k+7

    Recall that: 9k - 2k = 7k

    So, we have:

    9(k+1) - 2(k+1) = 7k+7

    Factorize

    9(k+1) - 2(k+1) = 7(k+1)

    Since the above mathematical statement is true, then the given statement has been proved

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