Question

to prove 7^n+2+8^2n+1 is divisible by 57 using mathematical induction

Answer 1

Step-by-step explanation:

Say statement P(n) states 7^(n + 2) + 8^(2n + 1) is divisible by 57.

Check whether P(n) is true for n = 1:

=> P(1) = 7^(1 + 2) +8^(2(1) + 1)

= 343 + 512

= 855, which is divisible by 57

Now, let P(n) be true for n = m, means,

Let 7^(m + 2) + 8^(2m + 1) is divisible by 57.

Say, 7^(m + 2) + 8^(2m + 1) = 57A, A is any real positive integer.

=> 7^(m + 2) = 57A - 8^(2m + 1) ...(1)

For n = m + 1,

=> 7^(m+1 +2) + 8^(2(m+1) +1)

=> 7^(m + 3) + 8^(2m + 3)

=> 7.7^(m + 2) + 8².8^(2m + 1)

Using (1): substitute 7^(m + 2)

=> 7[57A - 8^(2m + 1)] + 64.8^(2m + 1)

=> 7.57A - 7.8^(2m + 1) + 64.8^(2m + 1)

=> 7.57A + (-7 + 64).8^(2m + 1)

=> 7.57A + 57A.8^(2m + 1)

=> 57A[7 + 8^(2m + 1)]

As 57 is out of the braces, 57 is a multiple of P(m + 1). In short, P(m + 1) is also divisible by 57.

Proved.

Answer 2

Given: 7^(n+2)+8^(2n+1) is divisible by 57.

To find: Proof by mathematical induction

Solution: Let P(n) = 7^(n+2) + 8^(2n+1).

By mathematical induction, first we need prove that P(n) is divisible by 57 for n = 0.

So P(0) = 7⁽⁰⁺²⁾+ 8⁽²ˣ⁰⁺¹⁾ = 7² + 8¹ = 49 + 8 = 57.

57 is divisble by 57, so P(0) is divisble by 57.

Now, for n = 1.

P(1) = 7⁽¹⁺²⁾ +8⁽²ˣ¹⁺¹⁾ = 7³ + 8³ = 343 + 512 = 855.

855 is divisble by 57, so P(1) is divisble by 57.

Now, let P(n) be true for n = m.

Hence, 7^(m + 2) + 8^(2m + 1) is divisible by 57.

7^(m + 2) + 8^(2m + 1) = 57k, where k is any real positive integer (say)

=> 7^(m + 2) = 57k - 8^(2m + 1)

For n = (m + 1),

P(m + 1) = 7^(m+1 +2) + 8^(2(m+1) +1)

= 7^(m + 3) + 8^(2m + 3)

= 7.7^(m + 2) + 8².8^(2m + 1)

Substituting 7^(m + 2) in the above equation,

P(m + 1) = 7[57k - 8^(2m + 1)] + 64.8^(2m + 1)

= 7.57k - 7.8^(2m + 1) + 64.8^(2m + 1)

= 7.57k + 57k.8^(2m + 1)

= 57k[7 + 8^(2m + 1)]

Hence, we see that P(m + 1) is divisible by 57.

Therefore, we have proved that 7^n+2+8^2n+1 is divisible by 57 using mathematical induction.