17 Let n = 11 1 Find n + 1 * n + 1

Q: (-1)" n- In n

A: We have to use alternative test here.

Q: 8. For any positive integer n, (i+1)·2' = n-2"+1. [PROOF BY MATHEMATICAL INDUCTION]

A: Case 1 : n=1 LHS = ∑i=11(i+1).(2i)= (1+1).(21)= (2)(2)= 4RHS = (1).(21+1)= (1)(22)= (1)(4)= 4 Hence…

Q: n 21. if an+1 = an an 2n + 3 n=1

A: We have, an+1=n2n+3an Substitute value of n=0,1,2,3,4,..... At n=0, a1=020+3a0a1=0 At n=1,…

Q: Prove n n(n + 1)(2n + 1) Σε ん? 6 k=1 for every n E N.

A: Given ∑k2k=1n=nn+12n+16, n∈N

Q: 10. 1²+2²+...+ n² = n(n+1)(2n + 1) 6 integer n ≥ 1. s 12 for every

A:

Q: Prove that (1² × 2!) + (2² × 3!) + …· + (n² × (n + 1)!) – 2 = (n+ 2)! (n – 1) ... For all n E N

A:

Q: n°(n+1)² 1 +2° +... + n³ %3D for every natural number n. 4

A:

Q: 1 Answer: diverges n In n n=2

A: If the integral ∫1∞fxdx=∞ then ∑i=1∞an diverges.

Q: (-1)" /n + I is n + 2 n=1

A:

Q: सिद्ध है कि n! > 3 ^ n सभी पूर्णांकों के लिए n> = 7।

A: हमें दी गई समस्या को हल करना है:

Q: 00 In(n) . Σ n=2

A: According to question given that ∑n=2∞ln(n)n

Q: For every positive integer n     12+22+32+...+n2=n(n+1)(2n+1)/6 Let S(n)=12+22+32+...+n2 How do…

A: See the attachment

Q: a. In n In(n+1) {")-1} = {"}

A:

Q: 1 For all integers n21:+ 1.2 2-3 3. n(n +1) n+1

A: Calculate the left hand side of the given equation. 11·2=12 Calculate the right hand side of the…

Q: 2+1/n 1+1/n 2n + 1 2 2 for n → o. 1 In n+1

A:

Q: 1 Let n = Find n + 1 and n + 1 n + 1 = %3D 1 %3D n + 1

A:

Q: Σ (-1) n=1 nyerges for all n20

A: Please refer to the image below

Q: 1 2. Let e =. Determine the smallest choice of N > 0 such that N. 6 ' n(n + 1)

A: To determine the smallest choice.

Q: (-1)" n In n n=2 5.

A:

Q: 29. For any odd integer n, n° n² +3 %3D 4 4

A:

Q: Let n = WEE 5 Find n - 1. n – 1 =

A: Given:- n=-2/5 To find:- Value of n-1

Q: Prove 31(n3 – n), for n = 0,1, 2, 3, ... %3D

A:

Q: Use mathematical induction to show that, for any positive integer n, we n have that n(n+1)(2n+1) k2…

A:

Q: 7. For every integer n z 1, n(5n – 3) 1+6+11+16++(5n– 4) 2

A: Solution is in step 2

Q: 1 n In n n=1

A:

Q: 20 1 Find 13 Let n = n + 1 1 n + 1

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Q: 18. Prove: n ΣΚ k = 0 = n(n + 1)(2n + 1) 6

A: The given problem is to prove the given following identity that sum of squares of first n natural…

Q: Let n = 9. Find n - 1. n - 1 = a/5

A:

Q: Prove by Mathematical Induction that: 1∙1! +2∙2!+ +...+n∙n!= (n + 1)! –1

A:

Q: 3. Use principle of mathematical induction to prove that (n)(n + 1) 1+ 2+3+ ·..+n = for n > 1.

A: Solution

Q: 1. Prove: If n is odd then 5n + 4 is odd.

A:

Q: Q4) Prove for any natural number n 20 Li 1+2+2+. +2" = 2-1 ミ (n+1) .... (11) 1+2+3+ + n = 2

A: Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: 4. Use mathematical induction to prove that 1+2+3+..+n =;n(n+ 1)

A:

Q: Show by mathematical induction, that for all n 2 1 n! > 2n-1

A: To show → by mathematical induction that                       V n ≥ I  , n I ≥ 2n-1Proof →Let ,…

Q: Prove Γ (n) = (n-1)! where n ∈ ℕ

A: Concept: The proof for this gamma integral will be done using mathematical induction

Q: For which natural numbers n is n^3-10n^2+28n-19 a pri For which natural numbers n is…

A:

Q: where n is a natural number. i(1+1) n+1

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Q: 4 + 10 + 18 + . . . . . + n (n + 3) =  [n (n + 1) (n +5)] / 3  for n ϵ Z+       (Use Mathematical…

A: The solution is given as

Q: Q6 Use mathematical induction to show that n > 2n +1 for all integers n > 2.

A:

Q: 8. Use mathematical induction to prove that n 1 n k(k + 1) k=1 n+1

A:

Q: ) 1+3+5+ ...+ (2n – 1) = n² for all n > 1.

A:

Q: 4. (a) Prove that for every n EN we have n(n +1)(n+2) 1(1+ 1) + 2(2 + 1) + 3(3+ 1) + . + n(n+ 1) 3=…

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Q: Find σ(n) if n = 2^k-1 (2^k - 3) where 2^k - 3 is prime

A:

Q: 9. Show that if n is a positive integer then n Σ k² ( x ) = n(n + 1)2²−² k k=0

A: Concept: Mathematical induction: Mathematical induction is a technique for proving that a statement,…

Q: 4.6.8. Prove that n3 + 1 +-n is an integer for any n E N. 112

A:

Q: 1 Let n = Find n + 1 and %3D n + 1 n + 1 = %3D 1 %3D n + 1

A:

Q: Suppose n e z. If n² is odd, then n is odd.

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Q: Show that if n is a positive integer then n(n +1)2"-2 k=0

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Q: (n+1)!–1 Prove that - + n + (n+1)! is true for all 2! 3! (n+1)! integers n > 1.

A: The answer is as follows: