Use induction to prove the following proposition: 1² +2²+3² +4² + . +n² n(n + 1) (2n + 1) 6 VnE I+

Q: (5) 7(3k – 1)² +26(3k – 1)–8

A: On simplifying the above expression,we get

Q: Let n EU = Z and p(n): n² + n is an even integer. a. Write the following statement: (Vn) p(n). b. Is…

A: (a) Here, the given statement is: For ∀n∈U=ℤ, p(n):n2+n is an even integer.

Q: Prove: there exist no integer a and b for which 21a + 30b = 1.

A: We can prove this by contradiction. Take the negation of the statement and check if it is true, if…

Q: Show that 3 · 4n + 51 is divisible by 3 and 9 for all positive integers n.

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Q: 10 a.) Completely simplify E ()2* i=0 b.) Expand (1 – r)® and simplify. c.) Expand (r +r-1)5 and…

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Q: (a) Prove that (-1+iv3)" + (-1– iv3)" has either the value 2"+1 or the value –2" if n is any integer…

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Q: Find a positive integer n such that 2n is a square and 3n is a cube.

A: Sol:- Let 2n be equal to x², where x is a positive integer. Then, n can be written as n = x²/2.…

Q: to 2A+7 n-1

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Q: Determine the correct expression for L{6u(t+2)+&e4t – 58(t) Oa. 5 8-4 O b. be-2 - 5e- 8-4 6e2s - 5e-…

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Q: 2. Choose the best answer. Choose the correct graph that sketches f(2) > 4 and g(z<S+ log, with the…

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Q: given 7(x-1)=2(3x+2) prove x=11

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Q: Chow that ॥२ + १n + 3 (nt) (2nt3)(2nt5) (nt 4) 36

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Q: Disprove the statement by giving a counterexample: For all integers m and n, if 2m + n is odd then m…

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Q: Use contradiction to prove:  if 3n + 2 is an odd integer, then n is odd.

A: consider 3n+2 as odd integer  and let  n is even  therefore, n=2k for some integer k therefore,…

Q: Show that 3 · 4n + 51 is divisible by 3 and 9 for all positive integers n.

A: Given , 3.4n+51 is divisible by 3 and 9 for all positive integers n''-1The statement 1 can be…

Q: Find the term of (a – 2b)3 that has b in it. |

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Q: . Use mathematical induction to show that n Σ r=1 (2r − 1)² = 1² + 3² +5² + + (2n − 1)² = = n(2n −…

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Q: Prove that n2+n  is  divisible by 2, where nN.

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Q: Use proof by contradiction to show that for all integers m, 7m+4 is not divisible by 7.

A: Recall the following fact about divisibility.

Q: Perform the indicated operationsSubtract 7a to the 2nd power+8a-15 from the sum of 14a-5 and 7a to…

A: We are required to subtract 7a^2+8a-15 from sum of 14a-5 and 7a^2-8.First of all, we will find the…

Q: Prove that: n² + 3n 2n² +1 1 2

A: The given problem is to prove the sequence tends to 1/2.Given n2+3n2n2+1.

Q: a)(6n2– 3n + 9) - (n² + 3n – 5) b) (15a – 3w + 7) – 2(2a – 9) + (3w – 15a)

A: a) Given:         6n2-3n+9-n2+3n-5       =6n2-3n+9-n2-3n+5 combine like terms…

Q: CS CamScannELAH (b) ven ünat 28 +pv3=(q+2v3, where p and q are integers, find the values of p and of…

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Q: Use proof by contradiction to show that the sum of two odd integers(k and l) is eve

A: An even number is a number that is divisible by two. All even numbers finish with one of the digits…

Q: |(62" + 52") de Find + C

A: To find ∫6x4+5x5dx