Example 9.7.1 showed that the following statement is true: For each integer n2 2, n(n – 1) („-2) - (equation 1). 2 Use this statement to justify the following. (:) - n + 3 n + 1 (n + 3)(n + 2) , for each integer n 2 -1. Solution: Let n be any integer with n 2 -1. Since n + 3 2 , we can substitute in place of n in equation 1 to obtain (:::) - (L XC )-). n + 3 n + 1 2 By simplifying and factoring the numerator on the right hand side of this equation we conclude (*::)-L n + 3 n + 1

Q: Prove that n2 > 2n + 1 for n ≥ 3. Show that the formula is true for n = 3 and then use step 2 of…

A: To show that n2 > 2n + 1 for n ≥ 3 using mathematical induction.

Q: Show that (2i – 1) (2n)! for every positive integer n!2" n. i=l

A: We have to solve given problem:

Q: n = n and n = 1. n Show that in

A: A factorial is the product of consecutive integers starting from 1. It is known that nr represents…

Q: Given: In(1+ x) (-1)" xn+1 _if |x| < 1 n + 1 n=0 (-1)"(n + 2) 2n+1(n + 1) Compute the exact value of…

A: Solving

Q: 1 8 Let M -4 13 Find formulas for the entries of M", where n is a positive integer.

A:

Q: (b) Show that k-1 (k+1) n n+1 for every n ≥ 1.

A:

Q: The solution of e² = - 11 has the form x + ily + 2nn) where n is an integer. What is the value of x?…

A: The given equation is                                                ez=-11

Q: Prove that 2" < n! for any integern 2 4

A: We’ll answer the first question since the exact one wasn’t specified. Please submit a new question…

Q: Assume that S = Σ(-1)"-¹. 999 9 99 8 O 1000 10 1 (n + 1)5 n=1 n=1 Find the smallest integer N such…

A:

Q: Question 3. In parts (a) - (c), prove that for all n ɛ N: (a) n-1 £(:)-I- n k k=1 k=0 (b) £(:)-- n =…

A: Here we will use binomial series expansion  a+xn=n0an+n1an-1x+n2an-2x2+......+nn-1axn-1+nnxn…

Q: QUESTION 9 Use Mathematical induction to prove that 2 divides n2+5n, n21.

A: Question 9: Let n=1 is true for Let the result be true for n=k,  We need to prove the result is…

Q: Prove that 1-21 +2-22 + ……+n· 2" = (n – 1)2"+1+2 for any n E N. -

A: NOTE: According to guideline answer of first question can be given, for other please ask in a…

Q: Given: In(1+x) = >) (-1)". -xn+1 if |x| < 1. n+1 n=0 Compute the exact value of -1)"(n+2) 2n+1(n +…

A: Given: ln(1+x)=∑n=0∞(-1)nn+1xn+1 where x&lt;1To Compute: Exact value of ∑n=0∞(-1)n(n+2)2n+1(n+1)

Q: 1.Solve the following integer LP using branch and bound method: + 5 x2 MAX Z= subject to: + 10 < 20…

A: Answer

Q: The sum of any two integers of the from 4k + 1is always: Even Odd prime Zero Windows b

A:

Q: (-1)"(n + 2) 2n+1(n + 1) 4. Compute the exact value of n=0

A:

Q: Show that for any 1 ≤k≤n. T (1) = n²(n²-1²)...(n² (k − 1)²) 1.3.5 (2k-1)

A:

Q: Theorem 2.9. Let n be a positive integer and write it as n = N²m, where m is square-free. If m…

A: n be a postive Integer and wollte it as mn = N ^ 2 * m where m is square furee. fm Contains no…

Q: 6.20. For which integers n > 2 does there exist a o e A, such that |o|> n?

A:

Q: 4. Let bo = 1, b1 = 3, and for each integer k > 2, we have br = 4bk-1 – 4bg–2 %3D Prove that for all…

A:

Q: Proceed as in Example 3 in Section 6.1 to rewrite the głven expression using a single power serles…

A: Given ∑n=2∞nn−1cnxn−2+∑n=0∞cnxn+2 We have to rewrite the given expression using a single power…

Q: Prove that 12 - 2²+ 32 - ... .. + (-1)a-1n²=(-1)n-1n(n+1)/2

A: The given problem can be solved using mathematical induction:

Q: 9. (Chapter 10, #20) Prove that for every n E N, (1+2+3+ · ..+n)2 = 13 + 23 + 33 + ...+ n³.

A: To prove the result ''∀n∈N,1+2+3+...+n2=13+23+33+...+n3'' by Mathematical induction: First prove for…

Q: Show that for any n E N, n(n + 1)(2n + 1) k=0

A: The solution is given as

Q: (-1)"(n+ 2) 2n+1(n + 1) 4. Compute the exact value of n=0

A: Given series is  ∑n=0∞-1nn+22n+1n+1

Q: Question 2 Use mathematical induction to prove that n(n+ 1)(2n + 1) 6 i=1 for all integers n > 1.

A: ∑i=1ni2=nn+12n+16 For all n≥1Prove the above statement using mathematical induction

Q: Problem 5.8 Prove, for all positive integers n, 1 3 2n – 1 1 < V3n 2 4 2n

A:

Q: *3. Prove that l' + 2 + .. +n =n(n+ 1(2n + 1) for all ne N.

A: By using induction.

Q: numerically greatest

A:

Q: Proceed as in Example 3 in Section 6.1 to rewrite the given expression using a single power n(n –…

A:

Q: Show that 2n+1≤2^n  for all n≥3.

A:

Q: Prove n (2n = n n 1 k=1

A:

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

A:

Q: 10) Which of the following shows the solution for n2 + 2n 2 24? A) 4十 -6-5-4-3-2-10 123456

A: The given inequality is n2+2n≥24

Q: Show that 5n + 3 and 7n + 4 are relatively prime for all n.

A:

Q: Prove that if h> -1 then 1 +nh ≤ (1 + h)” for all non negative integers n

A:

Q: –1)" Given: In(1 + x) = > . if a < 1. n + 1 n=0 (-1)" (n +2) 2n+1(n +1) 4. Compute the exact value…

A:

Q: Problem 5.3 Prove that 5it = n(n+1)(2n + 1) (3n² + 3n – 1) 30 j=1 whenever n is a positive integer.

A:

Q: 8. (Chapter 10, #16) Prove that for every positive integer n, 2" +1< 3".

A: Using induction to prove any statement includes first proving the statement is true for n=1, then…

Q: Prove n 2 2n 1 k = n k - 1 n k=1

A: Binomial Theorem: According to the binomial theorem, let x and y are any two numbers. Then it is…

Q: Problem 4.2 Prove that for every positive integer n, n(n + 1)(n + 2) 1.2+2.3+ + n(n + 1) = ... 3

A:

Q: Let a1 = a2 = 1 and suppose an = an-1 + an-2 for n ≥ 3. Prove that a1a2 + a2a3 + · · · + a2n-1a2n =…

A: We have given that , a1 = a2 = 1 Also , an = an-1 + an-2 We need to prove that ,  a1a2 + a2a3 +…

Q: 1. Prove that if 7n2 + 5 is even, then n is odd

A: The given statement is,

Q: Show that 6n – 1 and 12n+1 are relatively prime for any n e Z.

A:

Q: -1)" =£ Given: In(1 + x) if x < 1. n + 1 n=0 (-1)" (n +2 2n+1(n + 1) 4. Compute the exact value of…

A: ln(1+x)=∑n=0∞(-1)nn+1xn+1     if x&lt;111-x=∑n=0∞xn  if x&lt;1

Q: 1. Show that 2n n=0 onverges absolutely.

A: Given series: According to the ratio test:

Q: QUESTION 3 Prove that 2n > n2 if n is an integer greater than 4. For the toolhar press AIT. E10 /RCL

A:

Q: Problem 27: Show that N when z = 1 X (z) = 1+z-+z-2 +...+z-(N-1) 1-z-N %3D when z + 1 1- z

A: We have to prove that the sum of the given series is equal to RHS.

Q: Question 13 Use Mathematical Induction to prove that for all integers n ≥ 1: (1 x 2) + (2 x 3) + (3…

A: