Use the binomial theorem to show that for all integers n 2 1, 3" = (6) + 2 (4) + 2° (") -

Q: (a) Suppose that n > 1 is an integer with prime factorization n = pf"p.p. Prove that (k, + 1)(k, +…

A: As both of them are different questions(not a subpart of other), so as per the guideline, I am…

Q: 4) Find the 5th term of (x+ y)°

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Q: Solve this attachment.

A: We have to show ∑k=0n-1knk=0 using binomial theorem. We know by binomial  theorem…

Q: 1. In your own words, what is the binomial theorem. Find the 5th term of (x+7)

A: Given : x+77

Q: Expand 3x + 2y)7 using the binomial theorem. Simplify your final result.

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Q: 6) Use Binomial expression in the form (a + b)¹: (x-2)5 + 10(x - 2)4 + 40(x - 2)³ + 80(x - 2)² +…

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Q: -4 M" = -127 14 Let M = 6 Find formulas for the entries of M", where n is a positive integer.

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Q: Find a formula for ¹ - 1²/ (²1) + 1² ( 2 ) -—- ... ₁ (2). (G) + (-) + I u

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Q: Use the Binomial Theorem to show that 50 E)(-2)" = 1. %3D

A: Binomial Theorem:

Q: 6. Use the binomial theorem to prove that 3 = C(n,0) + 2C(n,1)+2²C(n, 2) + +2"C(n, n)

A: By using the binomial thorem to prove that, 3n=Cn, 0+2Cn, 1+22Cn, 2+...+2nCn, n

Q: (1) How much is 0! ? (by convention on). 0!

A: (1) Given

Q: Prove that πd/n n = nτ(n)/2.

A: your question has a silly mistake, it should be πd|n d = nτ(n)/2. supposeτ(n)=r , where r be the…

Q: Write the binomial expansion for (x+3)°. ..... (x + 3)° =D

A: Solution...

Q: Show that when |z +1| < 1,z² =1+£(n +1)(z+1)" n=1

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Q: 11. Prove that every positive integer n can be expressed in the form n=4k or n=4k+1 or n=4k+2 or…

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Q: 3. Determine the 4th term of (3x+ 2y3 6 using The Binomial Theorem.

A: We will find the 4th term

Q: 9. (a) Using Pascal's triangle, expand and simplify (1 + 2)- (1-2.r)". (b) Using the binomial…

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Q: Find a formula for ¹ - 1²/ ( 7 ) + 3² ( 2 ) - ··· + + (−1)n. 2+1(1).

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Q: Prove the following using the binomial theorem: ()=()+ 2n

A: In this context, the aim is to prove 2n2=2n2+n2.

Q: 2021 2. Expand (1 + v3i)“ recall the binomial theorem.

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Q: 4.-Use the binomial theorem to find the first four terms of development 1 (4 + x)2

A: We have to find the first four terms of the expression (4+x)−12, we'll first rewrite it…

Q: *prove that for . P+2+3+... +n° = 2 n=!,2,3,

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Q: Which of the following is the value of the coefficient y on the right side of the equation given…

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Q: 3x2 Q1/ A/ Expand with binomial formula: (1- * Q1/B/ Find the value of G? -15 3- 36 = [4 -6 4 -13 -4…

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Q: 9. Let p be an odd prime, and let ( - 1)(a – 2).. (a - (p - 1)) = "--g12-2 -3 -...+ ap-32 - p-2a +…

A: Let a be any solution of x^2 + 1 ≡ 0 (mod p), so that a^2 ≡ −1 (mod p). Becausep doesn't divide a.…

Q: Use the Binomial Theorem to prove that 4 (¹ + 5)^² 5n 53 25 8 25n' for n ≥ 1.

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Q: (Vx - V3)° 6.

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Q: Indicate the least integer n such that f(x) = (3æ³ + 2x²)(5æ5 + 4æ*) € O(x").

A: Finding f(x) explicitly using multiplication of polynomial. The highest power of x or degree of f…

Q: M = 4 3 7 Find formulas for the entries of M", where n is a positive integer.

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Q: Factor 10+ 4i into product of primes Z [i], (Hint: Ille= (2)(2)(29)) in %3D

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Q: (2x-3y) =

A: