Use Gauss’s trick to determine H(n) = 1 + 5 + 9 + 13 + … + (4n – 7) + (4n – 3) + (4n + 1) I'm stuggling to know how many to multiply the sum by 2H(n) = (4n+2) (?) Is it n or n+1 or 4n+1

Q: 3-n (a) - 3n n=1

A: According to guidelines we solve one question.thanku

Q: 13 e, thank you.

A: e)Given that, sum of the first n terms of the given series is

Q: K CK-1) In(k-1) k-3

A: To find if ∑k=3∞kk-12lnk-1 is convergent or divergent: ∑k=3∞kk-12lnk-1 is convergent. To show this…

Q: Σ The coefficient recursion relation for the solution y An xn-1 of the n=0 equation xy" + 3xy' +…

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Q: 8. 3n+1 .251-n n#1

A: We have to solve given problem:

Q: The coefficient of x2 in the expansion of [1- x/2]3  -   (a + x) 4 is 2a where a>0. Find a.

A: The coefficient of x2 in the expansion of [1- x/2]3  -   (a + x) 4 is 2a where a>0. Find a

Q: [1 2 -3] (b) If A =0 3 0 2 -2 2, find the value of A" using Cayley Hamilton Theorem. Hence verify…

A: We will find out the required value.

Q: 3. The fifth term in the expansion of (2x + k) is equal to the fourth term in the expansion of (2 +…

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Q: (n - 11n +17n + 16) divided by (n– 3) n2 - 8n-6, remainder -6 n- 8n - 7, remainder |

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Q: 16) k(n) = -2|-3n|; Find k(3 + n)

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Q: How many terms will be in the expansion of (x – 8y)*" ? There will be 10 – 1 = 9 terms in the…

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Q: express     n-1 Σ((k^2)/n) k=1   in closed form.

A: The given question is related with closed form of finite sum. Given that,…

Q: A population grows according to the recursive rule PN = 4PN-1 , with initial population P0 = 5. (a)…

A: Given Information:  PN = 4PN-1 P0 = 5.

Q: Write out the sum. n- 1 1 Σ 5k+1 k = 0 Find the first, second, third and last terms of the su n- 1 1…

A: Q 1)- ∑k=0n-11/5(k+1)   Q 2)-∑k=0n1/9k

Q: 3. a) Obtain the expansion for (3 + 2x)* in ascending powers of x b) Find the term independent of x:…

A: a)

Q: a) Obtain the expansion for (3 + 2x)* in ascending powers of x b) Find the term independent of x: 3…

A: To answer the given questions we shall use the following identities 1.   (a+b)n=∑k=0nnkakbn-k 2.…

Q: c) n° n° + 2n is a multiple 3 for all n 2 1.

A: To show: n3+2n is a multiple 3 for all n≥1. Statement used: Principal of Mathematical Induction:…

Q: .3.4 .45 Find the sum. - 1- 13- 169- 2197 - ...- (13)" Sn = (Simplify your answer.) %D

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Q: Prove by the principle of mathematical induction that 3n2 + 3n + 1 < 2n for n ≥ 8 Ensure that you…

A: Given 3n2+3n+1&lt;2n for n≥8

Q: 3 divides n^3 + 2n, whenever n is a positive integer. Solve using mathematical induction

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Q: 12 Find the power of x in that term of the expansion of 2+ 2 which has the greatest numerical…

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Q: sequence is defined recursively as sk = sk-1 + 3k  for k ≥ 2. s1 = 4 s2 =  s3 =  s4 =  s5 =

A: A sequence is defined recursively as sk = sk-1 + 3k  for k ≥ 2. s1 = 4

Q: Q.No.2 State and prove Binomial theorem and also find 18th term in the expansion of (2x - 3y)3.

A: Given: To be prove binomial theorem and find the 18th term in the expansion of 2x2-3y25

Q: (a) Complete k (-1)* 2k+n = -½(x) k! (k + n + 1)! k=1

A: Solution a: Given ∑k=1k(-1)kk!(k+n+1)!x22k+n=-J?(x)            (1) Since we know that…

Q: Give a combinatorial proof (an algebraic proof will not be accepted ()()-(")- (;) 2n n 2 . -2. 3 3

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Q: 2 16 Use the binomial formula to find the coefficient of the y z term in the expansion of (2y-z)"°.

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Q: Please help I’m stuck on this one, I’m stuck

A: Introduction: Big-O notation analysis on polynomial function: If f(n)=a0+a1n1+a2n2+...+amnm, then…

Q: '글 +2+글 +5 The sum +8+ +11 written in sigma notation is equal to 8 3k-2 A k=1 2 8 3k - 1 B k=1 8 3k…

A: The sum 1/2 +2+7/2 +5+13/2 +8+19/2 +11.

Q: Σ (-1)* True/False: We have = -1. (2k)! k=0

A: We know thatcos(x) = ∑k=0∞-1kx2k2k!

Q: HW Solve the fallowing first order 0E. -3= \-3+メー

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Q: Let, a1= 3 and for n ≥ 2, an = 2an-1 + 1, express an in terms of n.

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Q: The series k=1 3k (k² + 1)“ + 20

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Q: a) Obtain the expansion for (3 + 2x)* in ascending powers of x b) Find the term independent of x: 3…

A: “Since you have posted a question with multiple sub-parts, we will solve first three subparts for…

Q: H(n) = 0 if n is  less than or equal to 0  H(n) = 1 if   n=1 or n=2  H(n-1) + H(n-2) - H(n-3)  if…

A: The conditions for H(n) is given as,

Q: 2. the number of terms in the expansion of  ( (x+4y)^3(x-4)^3)^2 is? 3. The 5th term of from the…

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Q: 1. Show that the binomial coefficient 2"Cn is always an even number for all n > 0.

A: Here, the given binomial coefficient is: Cn2n To prove that it is always an even number for all…

Q: ) 1+3+5+ ...+ (2n – 1) = n² for all n > 1.

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Q: Find the least positive integer n for which 2n + 5n - n is a multiple of 1000.

A: To find the least positive integer n for which the expression 2n +5n-n is a multiple of 1000.

Q: The coefficient of in the expansion of 1 (x+ + 1)7 is the sum of n multinomials. What is n? 3 1 4

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Q: 10. Prove that the sum of n squares can be found as follows n(n + 1)(2n + 1) 12 + 22 + 32+...+n² : 6

A: I use  a3-b3= (a-b)(a2+ab+b2)

Q: x4. Prove (1-1) (1-1)… (1-급) n+1 for all n 2 2. 2n 22 32 X5 From the homework and examples vou have…

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Q: (n+1)! %3D Solve for n: = 30 (n-1)!

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Q: In the expansion of (4x -2/x)8 or (4x -2x^-1)8, find the following: a)the term containing x^6 b)the…

A: Given that - In the expansion of (4x-2/x)8 or (4x-2x^-1)8, find the following: a)the term containing…

Q: 10. Find term number 3 in expansion of: (4x + 2)^9. (Note, term number 1 is (4x)^9) the coefficient…

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Q: Please explain how the -20k is brought back into the formula, Thanks Solution c: Obtain the sum: R.=…

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Q: 3n – 5n is n→ o n° - 2n² +1 lim 3

A: It is given that, limn→∞3n3-5nn3-2n2+1.We have to evaluate it.

Q: 28. The expansion of the binomial (x +y)2a +5 has 18 terms. The value of a is

A: