oof. Assume that n is an odd integer. Then n 2k + 1 for some integer k. Then %3D 7n - 5 = 7(2k+1) - 5 = 14k + 7-5 = 14k + 2 = 2(7k + 1). Since 7k + 1 is an integer, 7n- 5 is even. q.e.d.

Q: For all integers n > 1, E(4i – 3) = 2n² – n i=1 [ In other words: For all integers n > 1, 1+5+9+ ..+…

A: The given series is ∑i=1n4i-3=2n2-n, n≥1 To prove the statement by mathematical induction, we will…

Q: D. Show that if a, = 3n³ - 17n² + 2n - 6, then an O(n³).

A:

Q: 8n and IfΣ (ακ+ be) - n+1 k-1 2(ak - E(ak - br) = - n, then ak = n+2 k-1 k-1 3 O 2 9. m/2 2. wwwwww

A: simplify the equation by adding them and tends to limit to infinity.

Q: If Sn denotes the sum of the first n terms of an AP prove that S30 = 3 (S20-S10)

A:

Q: 5. For k(x) = 3 × - 6, determine kª(8) -

A: To find the required value we need to find the value of x for which k(x)= 8. 4x/5 -6 = 8 4x= 70…

Q: 2. Find p and q such that 7x+1 5 |5x-1|≤4 implies p ≤: <q

A:

Q: The statement 2n< n! is valid for For only n=4 All positive integers All positive integers n 2 4 All…

A:

Q: 2. Let an = 3n + 2 for n = 0, 1, 2, ... (i ) Find ao, a1 and a2. (ii ) Show that a2 = 4a1 – 3ao.…

A: Answer and explanation is given below..

Q: If p times tne pth term of an A.P. is equal to q times the qth term of the same AP., then how that…

A: Given that If p times the pth term of an A. P. is equal to q times the qth term of same A. P.  To…

Q: If (ar + br) 4n and n+1 k-1 E(ak - br) 1 then ar = | n+2 k-1 k-1 3 5. None 2. 72n/2

A: use elimination method Find limit for n tending to infinity.

Q: Prove that the sum of the first n cubes is equal to the square of the sum of the first n positive…

A: According to the given information, it is required to prove the following.

Q: Find a closed formula for a, if ak = 5n² + 4n k=1 an =

A:

Q: in class of N students where girls and boys and boys ratio is p:q savita ranked rth from the top. if…

A: Total number of students in class= N Ratio of girls to boys: p:q And it is given that Savita ranked…

Q: While proving- a(a+1)(2a+1) . is an integer for the case a = 6q +2 which of the 6 %3D following will…

A: Option (c) is correct. See the attachment

Q: If (n + 1)(n + 2)an+1x" - nan-1x" = 0, n=1 show that for k = 1, 2, 3, ..., we have 1.3.5... (2k - 1)…

A:

Q: Σ (-1) n=1 nyerges for all n20

A: Please refer to the image below

Q: Σ 9n n = 1 Identify an-

A:

Q: for all integers n >1 E, i = n(n+1} %3D 2

A: To prove the sum of n natural numbers n≥1 is  ∑i=1ni=n(n+1)2

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

A:

Q: Show that n*-n3.5+300n + 234n²log n³ + nVn10 e 0(n*)

A:

Q: For each positive integer n 2 0, let P(n) be the formula 3n+1 – 1 1+3 + 32 + ... 3n 2

A: We prove this formulas by using PMI

Q: Fix k, n E N with n > 2 and k <n – 2. Prove that: | )-(".) · ) (:) ) (^) = (":") 2 + k 2 n – 2 n n +…

A:

Q: For how many positive integers 'n', n3-8n2 + 20n - 13 is a prime?

A: Introduction :

Q: that Edn H(n/d)7(d) = 1 and Edn H(n/d)o(d) = n for all n, where r(n) = d(n) and σ(n) = σι (n) Σm d.…

A:

Q: Use the permutation and combination formulas to prove the following. (a) For all integers n 2 2,…

A: We will solve the problem

Q: . Show that if an = 3n3 - 17n2 + 2n - 6, then an = Θ(n3).

A:

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: (Vn)(n2 +n+ 17 is prime)

A: Since, you have posted multiple questions , I am doing the very question (ques 7)  according to our…

Q: While proving n+2)2n+2) is an integer for the case n = 6q + 5 which of the following will be…

A: Given that  n(n+1)(2n+1)/6 Plug n=6q+5

Q: Show that, for n > 1 2n 1.3.5... (2n – 1),2n 2.4.6...2n n

A:

Q: (4b) Let n E Z. Prove that if n is even, then n² + n + 2 is even.

A:

Q: [Hint: For m > 2, ( 2 7. For n > 1, verify that + 1 1² + 3² + 5² + ... + (2n – 1)° = ( 2"" ) - 8.…

A: Let's say we need to prove a formula or a statement P(n). To prove this, we need to prove that The…

Q: The sum of n,2n,3n terms of an AP are S1,S2,S3 respectively ,Prove that S3=3(S2-S1)

A:

Q: (-1)" (n+ 2) 2n+1(n + 1) Compute the exact value of

A:

Q: Which of the following sequeces in GP will have common ratio 3, where n is an Integer? a) gn = 2n2…

A:

Q: that d | n + 5 and d | n2 + 7, then d | 32. , then n + 5 and n2 + 7 are coprime.

A: For part (a) We have to prove that d is an integer such that dn+5 and dn2+7, then d32. It is given…

Q: The number N of deer in a certain population t years after observation began is given by  N =…

A: Given: The number N of deer in a certain population t years is To find: The deer population after…

Q: Find the possible values of n. (n2 )" – 2n" + 1 = 0 (n +0,1)

A: We have  n2n-2nn+1=0 and n≠0,1 We can write n2n as n2n=n2n=nn2 So, nn2-2nn+1=0 Let nn=x x2-2x+1=0

Q: only Q2(8) 2. Show that following statements are correct: į. 4n+100=0(n) iii. n³ ± O(n²) v. n! =…

A:

Q: 11) Which of the following shows the solution for n2- 2n < 24? 十十 A) 6-54-3-2-10123456 + +++++++…

A:

Q: Given the set of multiples of 5 defined by 5Z:= {m: m = 5n for all n e Z} together with the…

A:

Q: Suppose the final score in a hockey game is 4 to 2. How many possible scores are there after the 1st…

A: Team one had 5 different possible score total at the end of the 2nd {0,1,2,3,4} Team two had 3…

Q: (4a) Let n E Z. Prove that if n is even, then -5n – 3 is odd.

A:

Q: For a populationN = 22,000 and p = 0.14 . Find the for hat p =0.32 for n = 50

A: The value of p hat is 0.32, the sample size n is 50 and the value of p is 0.14.

Q: show that the sum of this: pi/(3sqrt(3))=(∑∞ n=0 ((-1)n (1/2)3n+1))/(3n+1) + ∑∞ n=0  ((-1)n…

A:

Q: If n is an odd integer, show that n4 + 4n2 + 11 is of the form 16k.

A: Solution: The objective is to show that given equation is of the form 16k

Q: for n greater or equal to 1  proove that 2+4+6+...+2n=n^2+n by using mathematical induction

A:

Q: 6n_ and If (ar + bx) = n+1 k-1 , then 1 (ak – br) = 712 n+2 k-1 k-1

A: If ∑k=1nak+bk=6nn+1 .......................(1)  and  ∑k=1nak-bk=nn+2-1n ......................(2) we…

Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.