1. 1.1! +2.2! +3.3! + ... + n.n! = (n + 1)! - 1 for all n ≥ 1

Q: A sequence is said to be decreasing sequence if Vn > 1 'n+1°n =S .a .b O n+1 2n.cO n+1 n+1d0 n

A: Option (d) is correct.

Q: 1 + 1.2 1 1 n for all integers n>1 n+1 3. Prove that %3| 2.3 n(n+ 1)

A: To prove the given statement.

Q: alc and b|c, then ab|c if a and b are not relatively prime.

A: The given statement is FALSE. Given a|c and b|c then ab|c if a and b are not relatively prime. If…

Q: X4 X6 x8 x10 3. Y = x2 + 3! N Terms ** * ... 5! 16 1L +

A: Please rate and feel free to ask any query

Q: n(n+1)(2n+1) Prove: 12 + 22 + 3² + ... +n? for all positive integers, n.

A:

Q: n3 + 3 n5 n4 - Prove that 14 + 24 + ..n* = +. n for all n > 1. 30 -

A: Consider the provided question,

Q: 5s L |(s – 2)

A:

Q: 2. Show that 7-(4-3 k=0 (-1) k + 1 \k n+1

A:

Q: 4. Conclude that n2 n=1 ||

A:

Q: n (n + () A k=0 | B, = 0.

A:

Q: What is the summation from n = 1 to infinity, of  (3n+1) / (5n + 4) ?

A:

Q: Prove n (2k – 1) = n² - k=1 for every n EN.

A:

Q: Wht is the sum of this 8.1 - 3.1 x 2.1 + 0.4

A:

Q: (a) Let n EN. Determine the set of all integers which leave a remainder n+1, n+2, n+3 when divided…

A: PART A As it is given that 2n+1, 2n+3, 2n+5 are consecutive odd integers so they are pairwise…

Q: 4. Find the largest value of n such that 2"|16!.

A:

Q: Show that n 1 i2 i=1 <2 - n

A:

Q: Solve for n. k (3-n) %3D

A: b=k(3-n)pbp=k(3-n)

Q: Question 12 Which of the following is not correct? B n²(n+1)(n+2) 3 is 0 (n) for every integer n ≥ 3…

A: (A) n2n+1n+23 is θn4 for every integer n≥3     ∴n2n+1n+23=n2n1+1nn1+2n3=n431+1n1+2n=θn4 Hence option…

Q: Let n = WEE 5 Find n - 1. n – 1 =

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

Q: Indicate the smallest integer n such that (6x²+4x)³ f(x) = € 0(x¹). 2A-4x On=4 O n=2 O n=1 O n=6 O…

A: f(x) = (6x2 + 4x)3/(x4 - 4x)         = {2x(3x + 2)}3/{x(x3 - 4)}        = 8x3(3x + 2)3/{x(x3 - 4)}…

Q: 1. Compute the value of (), for n > 3.

A: Solution

Q: Determine whether (5) () = (;) (") n-3 (where 3 < k < n) is true or %3D false. O True O False

A: We have given that

Q: Show the solution that a,+1 < a, for each n E N (-1)" п! n=1

A: Find your answer below

Q: (d) Show that for all integers n > 1 S(n, 1) = 1 = S(n,n), then, using the above relation (or…

A: Given: n≥1 To find Sn,1=1=Sn,n and show that S4,2=7

Q: 1. The factorial notation n! is defined for a positive number n as n! %3D п (п — 1)(п - 2)...…

A: Questions: 1. The factorial notation n! is defined for a positive number n as, n!=nn-1n-2...3210. 2.…

Q: X ME n! 3 11

A:

Q: n(n+1) For all n E N, prove 1 + 2 + ··· + n = 2

A:

Q: 5. Find and prove a formula for P(n,n-2) for n2 3.

A:

Q: 3. Prove that 6|(9" - 3") for all n E Z+.

A:

Q: 3. If а, 1000". Find N E I, the set of positive integers, such that an+1 N). n!

A: Given an=1000nn!. We have to find the N∈I such that an+1&lt;an, n&gt;N. For n=1 a1=100011! For n=2…

Q: Prove that n Sk k=1 = n(n + 1)(2n + 1) 6 for all n € N.

A: To prove:We will use mathematical induction to prove this.Let