Prove that for all n E N, 1 1x 2 x 3 n(n + 3) 4(n + 1){n + 2 1 1 2 x 3 x 4 n(n + 1)(n+2)

Q: Prove that for all positive integers n, we have 6″ – (-4)″ Σ (1) 5k 2 k−1 k odd =

A: Expand 1+xn 1+xn=C0n(1)nx0+C1n(1)n-1x1+C2n(1)n-2x2+.......+Cnn(1)0xn=∑r=0nCrnxn Expand 1-xn…

Q: Prove that for all positive integers k and n, with k < n, n-1) k-1/ k 1 k ... k k- 1 k- 1

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Q: Prove by the principle of mathematical induction that 1 x 1!+2 x2! + 3 x 3! +…+ n x n!= (n+1)!—1…

A: to prove the result 1×1!+2×2!+3×3!+...+n×n!=n+1!-1 for all natural numbers n , proceed as follows…

Q: Prove that for all positive integers, 1 1 + 3.4 1 ... 1.2 2-3 n(n + 1) n+1'

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Q: let k >= 2 show that n= 5^k-1 satisfies the equation 4σ(n) = 5n-1

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Q: n Prove that + 2 + 22 +...+ 2" +...+ 2n = 3" for all integers n 2 0. * ^) xn-ryr and x=4=1.…

A: We have to prove given identity:

Q: ^n for all n>=6

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Q: Prove, for all positive integers n, 1 3 2n – 1 1 2 4 2n 3n

A: we can prove given identity by mathematical induction method: Algorithim: Obtained P(n) and…

Q: Prove or disprove that for all n > 1 п i2' < 2"+1n. i=1

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Q: Prove that k() = n(1) (n = 1) for integers n and k with 1 ≤ k ≤ n, using a: a) combinatorial proof;

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Q: Prove, for all positive integers n, 1 3 2n – 1 1 - 2 4 2n (3n

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Q: Prove that 2n &gt; n for all positive integers n.

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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: n Σ k (k!) = (n+ 1)! - 1. %3D k=1

A:

Q: Prove that 12- 22 + 32+ +(-1)"n = (-1)**'n(n+1)/2 for all n E N.

A: Given : Prove that  12−22+32+......+(−1)n+1n2=(−1)n+1n(n+1)2   for n∈ℕ

Q: (1.3) Show that n - n (n – k) . k + (k + 1) k k 1 - for any two positive integers k and n with k <…

A: Let's find.

Q: Find a positive integer m > 3 such that there are at least 3 integers n in Zm such that n2 = 1.

A: We have to find a positive integer m≥3 such that there are at least 3 integer n in Zm such that n2=1…

Q: If we want to prove by contradiction the statement: For any positive real number x there exists a…

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Q: Use induction to show that 1 1+ 3 1 1 + 3n 3 1 3n+1 1 %3D 32 for all positive integers n.

A:

Q: 2) Let k be any integer greater than 1. Prove that for all n 20, we have: 1 (k – 1)· k +k' +k +...+…

A: Let k be any integer greater than 1. To prove that for all n ≥ 0 , we have:  (k-1)*[k0 +k1 +k2…

Q: . Prove that 2 +4+6+•…+ 2n = n(n + 1) for all n > 1.

A: According to the given information it is required to prove

Q: prove that (1- (1/4))(1-(1/9))(1-(1/16))...(1-(1/ n2)) =(n+1)/2n for all integers n greater than or…

A: We have to prove that  (1- (1/4))(1-(1/9))(1-(1/16))...(1-(1/ n2)) =(n+1)/2 By mathematical…

Q: Prove that n3 + 2n is divisible by 3 for all n E N.

A: We prove that n3+2n is divisible by 3 for all natural numbers n.

Q: Prove that lor n {k=1 positive unteger every K2K = (n-1)2²+¹+2

A: answer is in next step.

Q: Prove that 3 | 4" – 1 where n E N.

A: To prove 3|4n -1  That is 3 divides 4n -1 for every natural number n.

Q: Prove that for all positive integers k and n, with k < n, (:) - () - (^) n k 1 k n 1 k k – 1 k – 1 k…

A: 1+kk-1+K+1k-1+   ... ...+n-1k-1 add and subtract 1 and write 1 as a combination of k objects taking…

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

A: By using induction.

Q: For all integers n>0, prove: 2n < (n+2)!

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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: Prove that for all positive integers k < n, the equality n n n =2n-k k k i=k holds.

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Q: (a) Prove that for all integer n 2 1, 3|n(n² + 11).

A: a.          In the given problem we have to prove that for all integer n≥1, 3|n(n2+11). To prove…

Q: 1. Prove that (fn+1fn+3 – fnfn+4)* = 4 for each positive integer n.

A: We need to prove this, fn+1fn+3-fnfn+42=4

Q: Prove that there exists an n ∈ N such that 7 | (3^n − 1).

A:

Q: 4. Prove that for all n E N η Σ( r=0 = = (3) Hint: Examine the coefficient of x² by expanding both…

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

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Q: Prove that gn = 5 · 3n + 7 · 2n for all integers n ≥ 0.

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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: a) Prove that ε n(n+ 1)(2n + 1) Σ 6 i=0

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Q: n Prove that (3i – 2) = ; (3n – 1) for all integers n > 1. - i=1

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Q: Prove by any means that n!/(n − r)!r! is always an integer for 0 ≤ r ≤ n

A: Since nCr = n!/(n-r)!r! So given expression represents combination and it is read by "n chose r"…

Q: part ii and iii, thank you

A: According to pascal law,

Q: 1. Prove by mathematical induction that for all positive integral values of n, 2=1 =- (n + 1)(2n +…

A: (1)-According to question given that∑r=1n r2    ....(1)Now  apply the sum formula we get that∑k=1n…

Q: Prove the following: for all integers n > 5, n2 < 2"

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Q: Prove that n^2+5 >=4n for all integers, n>0.

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Q: Prove that the statements here are true for every positive integer n. 1 + 3 + 5 + · · · + ( 2n - 1)…

A: Show that for every positive integer n    1 + 3 + 5 + · · · + ( 2n - 1) = n2

Q: 1) use the mathematical induction to prove : A) 1x1! + 2 x 2! +3 x 3! + +nx n! = (n + 1)! – 1 %3D ..

A: note : as per our guidelines we are supposed to answer only one question. Kindly repost other…

Q: 2 Lk=0 4. Prove that for all positive integers n, п

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Q: 3. Prove that, Σk() = n2n-¹. k=1

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Q: 8

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Addition Rule of Probability

It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.

Expected Value

When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).

Probability Distributions

Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.

Basic Probability

The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.