Prove each of the following using induction: 1. Let a E R and a > 1. Prove a^ > a for all n E N. 2. Prove that 1/2 + 1/4 + 1/8 + ... + 1/2" = 1 - 1/2" for all n E N.

Q: 5) Prove that : tanz 올n(를) i-z

A: tan-1z=i ln[(i+z)/(i-z)]/2.

Q: 1), n = 1, 2, . .

A: In this question, the concept of mathematical induction is applied. Mathematical Induction Inductive…

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Q: 3" 7.

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Q: y. Use strong induction to prove that if n EN and n> 10

A: Base step

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Q: Prove that 2n+1 > (n + 2) · sin(n) for all positive integers n.

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Q: Prove by Strong Induction that, 2n-1+2n-2 = 3 x 2n-2 for every positive integer n.

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Q: 4. More weak induction Use weak induction on n to prove the following claim: n! 1

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Q: Use mathematical induction to prove и исин) {i= VEN 2 1 = 1

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Q: Show by example that Σ(an/bn) may diverge even though Σan and Σbn converge and no bn equals 0.

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Q: R² = gen gen

A: see below the reason

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A: We have to prove 2⋅3n−1≤2nn by using of principal of mathematical induction, where n≥0 Recall that…

Q: The Fibonaccit sequence Un} = 1, 1, 2, 3, 5, 8, 13, 2 1, ... is defined recursively by f1 = 1, f2 =…

A: Given: fn=fn+1+fn+2  for n=1,2,3, ...with f1=f2=1To prove: fn <2n ∀ n∈ℤ+Let for n=1, 21<2 and…

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Q: Establish formula for the sum of first n odd positive integers and prove it by using mathematical…

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Q: (n+1)(5n+6) Prove that 3+Č(3+5j)= j=l

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Q: Suppose that an<0 for all n, and define b,=tan-(a,-n). Then the sequence {b,}... O diverges to - O…

A: Solve the following

Q: of

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Q: Prove by induction that for any non-negative integer n > 4, 3^n < (n+1)!

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Q: Prove the following equation for n^2 >5n+2 for every integer n>=100 by mathematical induction

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Q: Let k be a positive integer. Prove ÷VE. Note: do not use induction. j=1

A: The solution are next step

Q: 4) The graphs of y=x and y=x make it obvious that x >x for all real number x > 1. Use induction (not…

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Q: prove its true by mathematical induction

A: The mathematical induction technique is used to prove the validity of the statement for every…

Q: ar – 5ak-1+ 6ak-2 = 5k, ao = 1, az = 3

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Q: How many bitstrings are there of length 2n that contain exactly n “O"s and n “1"s? Prove your result

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Q: Verify by mathematical induction Ž (2i – 1)° = n°(2n² – 1) for alln2 1. - i=1

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Q: (a) Let P(n) be the statement that: e-/ lim =0 for any positive integern • ()Use L'Hopital's rule to…

A: We’ll answer the first part of this question since due to complexity. Please submit the question…

Q: 5. Prove that . Hint: Compute e-' from the continued frac e - 1 1 + 2 + 3 + ·…. tion for et given in…

A: When a series is in the form ∑k=1∞-1k-1α1α2α3...αk, we can express it as a continued fraction as…

Q: 6. Use mathematical induction to show that, Vn > 7, n! > 3".

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Q: n>1 Problem 4. Prove by induction that, for every geq2 1 n +1 - - - 32 n2 2n

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Q: Ofind an upPer bound for the reciprocaL of the modulus of ż"-52+6 if 121=2 2Itint: z" -52²+6= (2?…

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Q: 5. Show that cos .3 G) do not converge.

A: We need to prove that: cosnπ3 does not converge This can be proved as follows:

Q: 1c ical induction to prove 11

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Q: 7) Show that if a, > 0, and Lim na, +0, then the series aj + az + az + a4 + diverges.,

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Q: Use strong induction to prove that if n EN and n > 10,

A: Base step

Q: Apply Mathematical Induction to prove that 3* > x² that for X>=1

A: first we will show that statement holds for initial value x=1 second we will assume that statement…

Q: Q12. Prove using mathematical induction and strong induction that every amount of postage of 12…

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Q: Let 0 < 0 < 1 be a rational number. Show that the sequence {cos(nT0)}-1 is divergent.

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