For n is greater than or equal to  1, use mathematical induction to establish each of the following divisibility (a) 8 divides 5^(2n)+7.  [Hint: 5^(2(k+1)) + 7 = 5^(2) (5^(2k) + 7) +(7 - 52 · 7)]   b) 15 divides 2^(4n) - 1

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Q: prove that is 11^k-4^k is divisible by 7    use mathematical induction

A: we have to show 11k - 4k  is divisible by 7 By using mathematical inductions.

Q: i) 3(-2) – (-2)7 + (-1)_ 2(5) – 6(-3) =

A: 3(-2)- (-2)*7+-1=-6+14-1=19 3(-2)- (-2)*7+-1-2(5)-6(-3)=-6+14-1-10+18=19+8=27

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Q: Value of the 7th term in (2x + y)"?

A: Follow the procedure given below.

Q: I was asked to prove that 5^(2n+1) + 4^(2n+1) is divisible by 9 for all n that are positive…

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Q: Find all integers n such that (n^2) - 1 is prime.

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Q: This will be (n+k-1)= 29+6-1 29 -¹1) = 34 29 = 278, 256.

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Q: For 7-10 Given a_1= 3, a_2= 7 and a_n= [2a] _(n-1) + a_(n-2) for n 23, 7) Find a_3 =,* Your answer…

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A: The solution is given below:

Q: What is the coefficient of x^8 in the expansion of (2+x)^11?

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Q: Let k be any positive integer. Use induction to provethat for every integer n ≥ 0, k^(n+2) + (k +…

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