Expand the following using the pascal's triangle 1. (5+ 2d)3

Q: 1. If the 2nd, 3rd and 4th terms in the expansion of (a +x)" are respectively 240, 720 and 1080.…

A: As per our company guidelines, we are supposed to answer only first question. Kindly repost other…

Q: stion 12 Suppose that In 2 = a and In 5 b. Use prop

A: Suppose ln2=a and ln5=b, Given expression is We will use the following properties of log to express…

Q: 5n(п + 1) 5+10+15+ .. + 5n %3D

A: To prove the given statement by mathematical induction

Q: (d) Given that n is even, how many 00-free bitstring of length n where n is even have exactly (끌 +…

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Q: 5. Let a = 2, az = 4, and an+2 a, = 2" for all natural number n. = 5a+1 - 6a, for all n 2 1. Use the…

A: Given information: The given values are a1=2,a2=4 and  an+2=5an+1-6an for all n≥1. To prove : an=2n…

Q: Problem 2. Let n EZ be such that 3 n. Prove that n' =n (mod 63). (Hint: 63 = 7-9, (7, 9) = 1, and…

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Q: Question 2 Prove the statement:: "For all integers n, if 3n+ 5 is odd, then n is even." (Hint: You…

A: The contrapositive of a statement is a proposition formed by contradicting both the  hypothesis and…

Q: 1. Give the direct proof of the following. a. If n is an odd integer, then n2 is odd.

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Q: (a) State and prove Pascal's Identity.

A: According to Bartleby's policy we can solve only 1 question at a time. Apologise for the…

Q: 1. Write out Pascal's Triangle through the 10th row.

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Q: Using a suitable form of Mathematical Induction to prove that 3" 7.

A: When we test the validity of a mathematical statement for all positive integers, we follow the…

Q: Prove the Pascal's identity when n 6. (")+?(")+ 2(")+ + 2"() = 30 en

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Q: Prove Pascal's identity, using the combinations formulas. Ck Ck + Ck-1 n+1 n

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Q: I was asked to prove that 5^(2n+1) + 4^(2n+1) is divisible by 9 for all n that are positive…

A: SOLUTION-

Q: Express in the form & j(n/2-j) (2) In(3

A: As per the bartleby guidelines for more than one part only one has to be solved. Please upload the…

Q: What are the first 5 entries in row 100 of Pascal's Triangle? O 100 C1, 100 C2, 100 C3, 100 C4, 100…

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Q: 1.5 Prove that le-2| 0.

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Q: ()() 6663 6665 1 Without using a calculator, prove that 2. 6664 6666 100

A: To prove 1234......6663666466656666≤1100 Now,…

Q: One of the foll owing is not true: Oa 0= 10 mod 5 b None Oc 14=2 mod 6 Od 20 =0 mod 5 Oe -24 mod 6

A: Here we use rules of modular operation.

Q: 9. Prove the formula n(n+1) Si=1+2+...+n= 2 i=1

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Q: Q: Apply Taylov's expansion and find the seyies for tenn at 4

A: Taylor series of a function is given by: ∑n=0∞f(n)(a).(x-a)nn!

Q: Notice that if we expand out 38 the following which are correct about the above proof. a. Actually…

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Q: Prove NIn, +oo[= 0 ! n=1

A: To prove that ∩n=1∞[n,∞)=∅. Suppose that x∈∩n=1∞[n,∞). Then, x∈[1,∞), x∈[2,∞),.... That is, x≥n,…

Q: 1. If the 2nd, 3rd and 4th terms in the expansion of (a + x)" are respectively 240, 720 and 1080.…

A: Topic:-  expansion function

Q: 4 O 1408 none of the above

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Q: 10. Prove the following identity 2 n+1 k=1

A: To Prove - ∑k = 1nn - 1k - 12n - 1k-1 = 2n + 1       Formula used - ab = a!b! a-b! where a! =…

Q: 7 Use mathematical induction fo derive the following formeld for gll nzi!

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Q: Prove bY inductions (R4

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Q: Use the formula to compute In(1.11) correct to five decimal places. In(1 – x) = - x" n = 1 In(1.11)…

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Q: 1 +2 +33 + ... +n ² = "°in + 1}°

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Q: 2 (i) Show that the result 1+ 5+9 +... + (4n – 3) = n(2n – 1) is true %3D for the casen= 1. (ii)…

A: The given question mainly depends on the the sum of the sequence which is in the form of arithmetic…

Q: Prove using mathematical induction.  1+3+5+⋯+(2n−1)=n^2, where n^2 means "n squared"

A: Principle of Mathematical Induction: Mathematical Induction is a technique of proving a statement,…

Q: (2i – 1) = 1+3 +5+7+ ... + (2n – 1) = n² i=1

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Q: irline sells n= 125 tickets for a flight that holds only 120 passengers. The robability that a…

A: A random variable X is called to follow the binomial distribution if it takes only non-negative…

Q: prove by mathematical induction + E+ 7.16 +.. 1 1 (Зп-2)(3п+1) 3n+1

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Q: Prove truth of formula below using mathematical induction. k(3k +7) 5+8+11+...+(5+3(k – 1)) =

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Q: |- Show Pascal's identity k 1 k k |

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Q: Claim: Vn E Z if n is odd then n2 is odd. (5.1) Give a direct proof of the above claim. (5.2) Give a…

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Q: 2. Use Fermat's Little Theorem, or its corollary, to find the units digit of 72018 + 112019 + 132020…

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Q: Let an be then th digit after the decimal point in 3T + 7e. Evaluate Ean(-1)". n=1

A: This is a problem related to sequence and series part. based on the general formula available for…

Q: In the question that follows, the "units digit" of a number is the last digit in decimal…

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Q: 6. No : o Show that sa= (Q-1)a

A: To show :- ∆ ax = a-1 ax

Q: Problem 1. Does there exist a natural number n such that 13 (n² + 1)? Prove your assertion.

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Q: Question 2. Prove that 2"32n -1 is divisible by 17 for all n E N. Hint. Induction.

A: Mathematical induction is used

Q: B. Prove each of the following by mathematical induction 3. 2k > 1+ k

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Q: Express the following in sigma notation 4+ 5+ 6+ ..+ 36 = > %3D i=4

A: Solution

Q: 7. Prove by induction, showing all of your algebra steps: 2.2¹ +3.22 +4.2³ +...+(n+1)2 = n. 2+1 for…

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Q: prove the following if 01

A: Suppose a∈ℝ such that 0<a<1 If possible, let us consider that a2>1 Now,…

Q: O Prove, by induction, that 3" +4" < 5ª for all natural numbers n, greater than or equal to the…

A: (a) The given inequality is: 3n+4n&lt;5n Substitute n=1 into the inequality and check whether the…