Exercise 2. Show that for any integer n the following inequality holds. (11) (1+> n+1 n+2 2n 2n +1 n+1 2n -1 2n Deduce e > 2. Show that (1+ )" > (1+5), and deduce that fan(z) 2 n(2).

Q: Verify that the following inequality is true for all integers n > 1 1 1.3.5. .... (2n 1) - 2n…

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Q: 4. Use the PMI to prove the following for all natural numbers n. |(c) E-1 2' 2n+1 – 2 i=D1

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Q: 10. Show that for all n E Z, n² – 3n +1 2.

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Q: (iv) It is known that 1+2+3+..+n== (n + 1) (Do not prove this). Use the formula above and the AM/GM…

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Q: 5. Prove by induction that for positive integers n, E-1(3i² + 2i + 5) = ÷(2n² + 5n² + 13n).

A: Given statement is : ∑i=1n(3i2+2i+5)=12(2n3+5n2+13n) Step1: Let us check if or not the given…

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A: Given statement is

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Q: Q2: Prove by Induction that Ŝ 2i+1 = h z0 ( The som of the fist had positive odd integers fum is…

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Q: - Prove that for all n E Z, we have ged(27n – 20, 4n – 3) = 1.

A: Solution

Q: Prove that for all positive integers n.

A: To prove for all positive integer n ∫01(lnx)ndx=(-1)nn!

Q: Exercise 3: Let n E N withn > 1. Show that (1 +x)" > 1+ nx for all x < 0.

A: Given that

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Q: M + N = {m+n : m € M and n € N}. Prove that for all n € N, the number sup(M + N) − n is an upper…

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Q: Q.No.6. Prove that by mathematically Induction 2n < (n+ 1)!, for all integers n 2.

A: (.)     2n &lt; (n+1)!      ∀integers  n≥2..................(*) (1)   now we check the above…

Q: Finish this proof by mathematical induction by completing the missing steps 3 and 4 Prove…

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Q: Prove the following inequality (1+z)" > 1+ nx + n(n 120 for any n2 1. Show that for all positive…

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Q: (e) Prove that for any non-negative integer n, i 3 n - 3"+1 – (n+ 1)3" + 1] 4 j=0 (f) Prove that for…

A: (e)…

Q: 2. Prove using induction that the following inequality holds for n2 0: (1+)"21+ n

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Q: п 6. Prove that E(8i – 5) = 4n² -n for every positive integer n. п i=1

A: # Given: sum(8i-5) =4n^2-n  i=1,2,3,.....n  We have to prove the above equation??

Q: 5. Prove that for all n n [j³ = j=1 n² (n + 1)² 4

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Q: (f) Prove by induction that for any positive integer n, n(n +1)(2n + 1) j=1

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Q: Q3/A/ Prove that (") + („",) = (*). n+: \k+1. n %3| k+1

A: Given:

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Q: Verify by mathematical induction Ž(2i-1 1. 1 - n° (2n? – 1) for all n 2 1. n(n+1) 2. Prove: V n € Z,

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Q: 17. prove that Edln H(n/d)7 and σ(n ) σ(n) Σ %3D

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Q: Show that for any n EN where n > 1, II (1 = n + 1. k k=1

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Q: Solve this inequality for n. 5 < 0.00001 0.4 n 5 s 0.00001n04 Multiply both sides by n° 4. 500000 s…

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Q: Q.No.6. Prove that by mathematically Induction 2n < (n + 1)!, for all integers n 2 2.

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Q: Prove that for every n E N is a² | (a + 1)^+1. - 1)n+¹ na — (a + 1), a € N. - -

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Q: Prove that the following inequality: n-1 Eklg k <n² Ig n 2 k=2

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Q: Q.No.6. Prove that by mathematically Induction 2" < (n + 1)!, for all integers n2 2.

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Q: Consider the sequence of real numbers defifined by the relations: X1 = 1 and xm+1 1+ 2xn for all n…

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Q: 2n° 1. Prove that for all positive integers n > 1, the inequality 2" < <(") holds. п

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