Calculate the limits of the real-number 3n+4 5n+6 an and bn = sequences (an)neN and (bn)neN, where (1+2n)(1+3n) e-n n³e-2n + n² e-n for n € N.

Q: If n be positive integer, show that the algebraic sum of the numerical coefficients in the expansion…

A: As given above that n be a positive integer. And the given expression is: 3x-2yn We have to prove…

Q: (n+1)! (n-1)! = 10n

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Q: (2n)! – (2n + 1)! (2n)!

A: It is required to find out , 2n! - 2n+1!2n!  For this we use factorial properties.

Q: Expand and simplify. (8h +3) (7h² – 4h+1) - O 56h3 – 32h² + 8h + 3 - O 56h3 – 11h? 4h + 3 O 56h3 +…

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Q: Write the binomial expansion for (x+ 2y)*. (x + 2y)* =D

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Q: How many strings of 1our decimal digits have all four digits that are different? How many strings of…

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Q: { :Find the first 3 terms for 2n-1 (2n-1)! (−1)n+¹

A: given sequence  -1n+1x2n-12n-1!

Q: Let a, = 1 and a₁ + a1 = n² + 2n, n ≥2. Evaluate S

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Q: . (a) Find the coefficient of r in the expansion of ‹ (² + ²)² (+ -)

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Q: The coefficient of x²y4z5 in the expansion of (3x+y+z)¹¹ is

A: The given expression is: (3x + y + z)11 To find: The coefficient of x2y4z5 in the expansion of (3x +…

Q: Which simplified choice will result in 200m20n15

A: Consider the following expression…

Q: "Co 34 "C, "C, "C; (-1)" "C, ,ne Nis equal to 10 14 (4n + 2) 2" n! 2"-n! () 3.5-7...(2n - 1)(2n +1)…

A: We have use binomial theorem  1-xn=nC0 - nC1 x + nC2 x2  + .........+(-1)n   nCn

Q: Find the coefficient of x3y2z3w in the expansion of(2x + 3y − 4z + w)9.

A: The given expansion is 2x+3y-4z+w9. We have to find the coefficient of x3y2z3w.

Q: What is the coefficient of z5y¹2 in the expansion of (1x + 2y)¹7?

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Q: (d) (n-r+1)! (n - r - 1)! *

A: According to the given information in this questionwe need to simplify the given form

Q: Solve for n if P(n,5)=120C(n,3)

A: The objective is to calculate the value of n, for which the following equation is…

Q: [1 2 3] ii) Let A = 4 5 6 and B = 5 L78 5] L3 of C32? [-5-5 71 [C11 C12 C13 - 2 4. Suppose BA = |C21…

A: First we state the general formula for multiplication of two matrices and then solve this particular…

Q: In the expansion of (x + y)16 which term has the same coefficient as x14y2? O a. x12,4 Oc x3y14 O d.…

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Q: Rewrite 1/2(x-5) +1/4(x-5)2 +1/6(x-5)3  +1/8(x-5)4 using summation notation.

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Q: If n/(a – b) and lcm(2b, 2n) = 2, then gcd(2a, 2n) = %3D %3D

A: We have to find the gcd of the two numbers   gcd= greatest common divisor the two numbers are 2a…

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Q: Which of the following is a false statement? In the expansion of (a +b) the coefficient of a b° is…

A: We have to choose the false statement.

Q: ´5s²–15s-11) -(s+1)(s–2)³. 34-2s 2. i) Find ii) Evaluate (15(s-5)(s+3).

A: The solution is given as

Q: Prove that product notation d/n = nτ(n)/2.

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Q: n² -1) 2n³ + 1

A: ∑n=1∞ -1nn22n3+1 use the alternating series test an=n22n3+1 an is positive and monotone decreasing…

Q: hich of the following expressions are equivalent to 2n 4n2 + 4n n+ 5 4n2 + 6n 10 (n+ 1) C 4n (n+1)…

A: given,

Q: 2 + 4 + 6+ . . . + 2n = n(n + 1)

A: On solving this we will get

Q: Simplify the expression Y = bar((AB + bar C) (bar (A + B) + C)) Obtain the canonical product of sum…

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Q: (b) 6 + 7 + 8 + 9 + ... + k

A: (b) Given that 6+7+8+9+...+k The objective is to find the sum. The formula for sum of numbers are,…

Q: If n(A) = 9, n(B) = 12, and n(ANB) = 5, what is n(AUB)?

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Q: Use the binomial theorem to prove that n <-D₁ (17) 3¹-*. 2″ = 2-D² k=0

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Q: 2(n-3)+n(2-n)=2-n2

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Q: O d. 3 34 9x + 3 Which of the following expressions does not represent the sum: Select one: O a. 2k…

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Q: A All of the above B(D-xXD+ x) = (D+xXD-x)

A: In this question, the concept of differentiation is applied. Differentiation Differentiation and…

Q: Use Gauss's trick to determine H(n)=1+5+9+13+...(4n-7)+(4n-3)+(4n+1) I got n(4n+2)/2 is this…

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Q: (c) How many strings of four decimal digits have exactly three digits that are 9? (d) Write down the…

A: We have to find the coefficient.