3. If the coefficients of 4 consecutive terms in the expansion of (1+ x)" are k1, k2, k3 and k4 respectively, then show that k1 k3 2k2 |3D ki + k2 ka+kA k2 + k3

Q: Find the term independent of x in the expansion of (2+ x²)°|1 2 3 1-). x2

A: Given- 2+x261-3x22 To Find- The term which is independent of x. Identity Used- For a general term of…

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: 20 Term independent of x in expansion of x2 5 + 6x3 is? e 7

A: Here we have to find the term independent of x.

Q: 16. The expansion of (2 + 3x;-1 is valid when >x>

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Q: Determine the coefficient of x3 in the binomial expansion of (1 + 5x)½ by completing the given…

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Q: The Lucas numbers, named after Franfois-Eduoard-Anatole Lucas are defined recursively by Ln =Ln-1 +…

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Q: 1. Solve the following recurrence relations (i.e., compute a closed form of In in term of n alone).…

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Q: (a) Find the binomial expansion of (1 – x)-1 up to and including the term in x?.

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Q: The first term of the expansion of (j – 3n)a is –-3nª. True or False?

A: The \displaystyle \left(r+1\right)\text{th}(r+1)th term of the binomial expansion of \displaystyle…

Q: S@ Find the coefficient f e in the expansion of (1-4x)2 for all non-negative integers n 6 Using…

A: Given that, (a) The expansion of 1-4x-12. Calculate the coefficients of xn. (b) Prove that…

Q: 5.2.8 suppose that a store offers gift certificates and denominations of $25 and $40. Determine the…

A: The given denominations are $25 and $40.Here, the given denominations are multiples of 5.The amounts…

Q: 6. Find the coefficient of x in the binomial expansion of 3. , if exists. 2.x?

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Q: Find the term with no x in the expansion of x 15

A: we have to find the term with no x in the expansion of x3+12x-215.  as we know that the (r+1)th term…

Q: Prove that for all integers n.4n² +6n + 1 is not divisible by 4. Be sure to explicitly show how each…

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Q: 1. the largest term in the expansion of ( 1+x)^19 when x= 1/2 is ? 2. If the 2nd,3rd and 4th terms…

A: (a) As we know that from the binomial expansion, (r+1)th and rth term is given by…

Q: 5 Find the coefficient of x in the expansion of x- The coefficient of x is

A: The general term of expansion of (a+b)n  Tr+1=Crnan-rbr Here, a=x, b=-2x, n=14 So, Tr+1=Cr…

Q: 4. Find the coefficient of the term containing x' in the expansion of (x-2y)“.

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Q: 2- (a) Equating the coefficients of xk+r-1 to zero Σα. +r)(k +r - 1)ax xk+r-2 - [(k +r)(k +r- 1) +…

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Q: 2- (a) Equating the coefficients of xk+r-1 to zero > {(k +r)(k + r- 1)ar xk+r-2 - [(k +r)(k +r - 1)…

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Q: Compute the coefficient of xy4 in the expansion of (3x – 4y)7. -

A: As per our guidelines, we can answer only one question in a single request. Kindly re-post the other…

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: 4 Which of the following is the correct expansion of (3x - 3y)2? 9 x2 - 18 xy + 9 y2 9 x2 - 9 y2 9…

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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: Solve the recurrence relation for computing the binomial coefficients (below): c(n, k) = c(n - 1, k…

A: The formula of combination : Crn =n!r!×(n-r)!           for n≥r To show Ckn = Ck-1n-1 + Ckn-1…

Q: 8 Use the binomial formula to find the coefficient of the zs° term in the expansion of (2z+s)".

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Q: #2. The binomial theorem states that for any real numbers a and b, In (a + b)n = =Σ" ("a²-kbk for…

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Q: 16. Find the term containing x18 in the binomial expansion of (x2- y)15.

A: A binomial expression is an expression of the form a+bn. This binomial expression can be expanded…

Q: 12. The first four terms of the binomial expansion of (1 - )' A I+ 4x + 7x² + 7x B 1- 4x – 7x + 7x³…

A: 1

Q: 4) Find the Term that free from x of the expansion (x2 +10

A: Use binomial expansion:-

Q: 6. Find the coefficient of x* in the binomial expansion of| 3 , if exists. 4 2.x

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Q: Obtain the expansion for (3 + 2x)* in ascending powers of x

A: According to binomial expansion,  a+bxn=an+n1an-1bx+n2an-2b2x2+n3an-3b3x3+....+nn-1abn-1xn-1+bnxn…

Q: 5. Find the generating function to determine the number of ways to pick k objects from 2n objects…

A: Given: When duplicates are allowed and the (2i – 1)th and (2i)th objects occur the same number of…

Q: Show that 5n + 3 and 7n + 4 are relatively prime for all n.

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Q: Find the first 4 terms in the expansion of (2 + x²)° in ascending powers of x.

A: Given expression: 2+x26  ……1   To find: First 4 terms (in increasing power of x) in the expansion of…

Q: If 2n + 1 is prime for some n ≥ 1, prove that n is a power of 2.

A: We have given 2n +1 is prime for some n ≥ 1 We have to prove that n is a power of 2.

Q: 2. Use the Binomial Theorem to find the first three terms in the expansion of (x + 2y)20.

A: We need to find first 3 terms of given expression.

Q: 3) Find the Term that includes x? of the expansion (x +

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Q: 2- (a) Equating the coefficients of xk+r-1 > {(k +r)(k +r- 1)ag x**r-2 - [(k +r)(k+r-1) +…

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Q: Which of the following shows the expansion of 2-5n? n-0 O(-18) + (-13) + (−8) + (−3) + 0 O(-3) +…

A: To find the expansion of the given expression, substitute n=0,1,2,3,4 in the given expression.

Q: 5. What is the coefficient of x''y" in the expansion (- 3x + 4y)ª . Show your work here:

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Q: Find the term independent of x in the expansion of 3 1 - 2r

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Q: Obtain the expansion of : 2 up to the term in x8 1-3x2

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Q: 6. Find, in simplest form, the middle term in the expansion of X* +

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Q: 4. The term in x* in the expansion of (1-x+x)1- 6. is: | .2 A. x* R 15 16 15 115 25 xD.x+ 4 - - 4 16…

A: As per the company guidelines we can solve first question. I hope that the given solution will help…

Q: 2- (a) Equating the coefficients of x*+r-1 to zero {(k +r)(k +r- 1)a, x**r-2 - [(k +r)(k +r- 1)…

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Q: Obtain the expansion of ; up to the term in x* 1-3x2

A: We know 11-x = 1+x+x2+x3+ ⋯

Q: A sequence is given by the recurrence relation Xn+1 = Xn + 3(n + 1) for n > 1, and the initial value…

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Q: Find the term of free x in the 10 2 expansion of (x² + 3)¹0.

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