Let S be the set of all real numbers except -1. Define * on S by a*b = a + b + a b. The solution of the equation 1*x*2 =5 is O a. 0 O b. 2 О с. -1 O d. None of these О е. 1

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A: we have the nth term of A.P is given by  an=a+n-1d here,  a=3, d=4-3=1an=3+n-11=2+n we get,…

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Q: Suppose thatk and n 2k are fixed positive integer. Justify the identity k+1 jk

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A: We have to use properties of union in this question.

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Q: Exercise 6 If az, a2, .,a, and by, b2, ..., b, are two sets of positive numbers arranged in…

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Q: 2. Show that for positive integers greater than 1, 2 + 6 + 10+...+(4n – 2) = 2n2 -

A: Given query is to find the sum of the sequence.

Q: let s be positive integer, then (2) + (;) = %3D

A: sC5 = s!/5!*(s-5!) And sC6 = s!/6!*(s-6)!

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Q: 3. If а, 1000". Find N E I, the set of positive integers, such that an+1 N). n!

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Q: 2. Show that if a]b then a'|b² for all integers a and b (Symbols V, 3, A, V, V,A,→,-)

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Q: Let Az = { k, k + 1, ..., k²}, for k 1,2,3, .. Find U은1Ak-nE1 Ax

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A: a)Here, the set consisting of the first 10 positive whole numbers.

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Q: - Show that (for n = 0, 1, 2, ..) (a) F(-n, b; c; 1) = - (c-b)n (c)n

A: note : As per our company guidelines we are supposed to answer ?️only the first question. Kindly…

Q: Suppose x is a fixed real number with x 2 -1. For n2 1, prove using mathematical induction: (1+ x)"…

A: Here we will show that given statement is true by mathematical induction method.

Q: 2 If A = 22 and B = -70 6. 10 , verify that B = s-AS, where s = -19

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Q: If det(A)=0, then:

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Q: Let a1,..., an be real numbers. Show that ajan + aja2+ aza3 +... + an-1ªn < i=1

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A: Solution: We are given that the set consisting of the first 12 positive whole numbers (i.e., 1–12)

Q: n Show that (") = (,“,) n-r.

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Q: Show that if fi(x) is O(g1(x)), f2(x) is O(92(x)), and f2(x) # 0 and g2(x) # 0 for all real numbers…

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Q: (b) Prove that 2m ≤ m! + 2 for m ¤ N. (c) For any nonzero real number x, prove ---3 i=1 = 1- x -n

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Q: (a) Determine whether the permutation 42135 of the set {1, 2, 3, 4, 5} is even or odd. (b) Find all…

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Q: Suppose that x is a negative real number and that x < 1/x. Prove that x <-1.

A: Since we know for any real number a,b  If a &lt; b then -b &gt; -a

Q: For any positive integer n and r = 1,2, ...,n, we have G) = (","') + C" - }). True O False

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Q: 4. For natural numbers n and k, prove (C) - ()

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Q: Let a and b be integers, not both zero. Then a and b are relativelyprimeif and only if a)1 = ax +…

A: Option D will be the right answer.

Q: 5. Deduce from (4) that the set {2} 2-: n = 1, 2, 3, ... is closed in R.

A: Given set  S =2 ∪2-1n  :  n=1  2,  3 , . . .

Q: Find the positive integers n such that , the sum of positive divisor oln) = 24,

A: Sum of Divisors: Let the prime factorization of n be n=p1a1×p2a2×...×pkak Then the sum of divisors…

Q: Show that for all positive integers n and k, (~^)= (-1)^(" + k-1). k

A: The provided information is  (  k-n) Then we prove this (   k-n)=(-n)*(-n-1)*(-n-2)*.....(-n-k+1)k!…