Prove that 2 + 2 · 22 + 3 · 23 + 4 · 2* + ...+ n2" = (n – 1)2n+1 + 2 for all n 2 1. Prove that 12 – 2 + 32 – 4? + ...+ (-1)"+1n² = (-1)n+1 n(n+1) for all n 2 1.

Q: Prove that 1+ 2 + 22 + 23 +. + 2" = 2n+1-1 for n 2 1. Use proof by induction.

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Q: If n is a positive integer with (n + 1)(n + 3) is odd, then (n + 2)(n+ 4) must be a multiple of 1) 6…

A: Given:  (n+1)(n+3) is odd We will get an odd number from product of two numbers only if the two…

Q: 4-(3n+2) = 4(- 3n+5) %3D

A: Given : 4 - 3n + 2 = 4-3n + 5

Q: Juan’s exercise regimen includes golf, swimming, and playing the drums. He has at most 12 hours per…

A: We have given all the important statement. We have to construct the linear programming problem.

Q: (n-1)! Use Euler's theorem to prove that if n is an odd integer, then n divides 2 -1.

A:

Q: Prove by Mathematical Induction that: 192^3+2364+ ...+n[n+1)(n+2) = n(n+1)(n+ 2){n+3)/4

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Q: Prove 0 + 1 + 2 + 3 + ...+ n = (n(n + 1))/2

A:

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

A:

Q: +6n +3 is Θ (n)

A: 1st expression is TRUE

Q: Using mathematical induction, prove that for all nonnegative integers n 2n+1 + (-1)" 3. 2" %3D j=0

A: For the solution of the problem follow the next steps.

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: 3) (n² + 10n+18) ÷ (n+ 5)

A: To divide the following expression.n2+10n+18÷n+5

Q: Find 은 dz V3z" 1 2z – 1 O a. (3r² + 3)+6(3z² + 2x–1)/ 3z/# /7² +3(6z+2) Ob. 2 + 6(3x² + 2x – 1) V3z"…

A: It is known that fg'=f '·g-g'·fg2.

Q: Use induction to prove that 2*3^(n-1) is less than or equal to 2n choose n (picture attached)

A: We have to prove 2⋅3n−1≤2nn by using of principal of mathematical induction, where n≥0 Recall that…

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: 2n+1+(-1)" Prove that for all integers n 2 0, 3.2"

A:

Q: how that 1+2+2²+23+...+2 = 2n+1 -1 for n≥ 0.

A: We need to prove by induction hypothesis      1+2+22+.........+2n=2n+1-1, for n≥0

Q: 1.-Proof by mathematical induction that 5 + 9 + 13 + ··· + (4n + 1) = n(2n + 3) ...

A:

Q: this

A: We use definition of t(n).

Q: 5+2-5+p 13) 3. 3) 28

A: Given: The equation 53p+23=5+p2p, To determine: The value of p,

Q: If n is a positive integer, then prove that 2n(2n-1) 2n(2n-1)(2n-2) 2! 3! 1-2n+- + ...

A:

Q: Prove for any natural number n that 12 + 22 + 32 + ...+ n² = ÷n 1 (n + 1)(2n + 1).

A: Step 1: This can be proved by using Induction method. This method involves following steps. Prove…

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 2+4+8...+2n=2n+1-2

A:

Q: If n is a positive integer, then (n + 2)! A = |(n + 1)! (n + 2)! (n+ 3)! is equal to (n+ 2)! (n +…

A: We need to find the determinant of the given matrix. To find the determinant we will use the…

Q: Find L[cosh 7t] s2-49 7 s2+49 s2+49 7 2 O O O C

A:

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: Va +3b + Va – 3b If x=- Va+36 – Va– 3b prove that 3bx² – 2ax +3b = 0 | - -

A:

Q: Prove that 1 + 3 + 5 +···+ (2n − 1) = n2 for every positive integer n (a) by adding 1 + 3 + 5 +···+…

A:

Q: (1) Prove by induction: Σ₁(i² − i + 1) = n³+2n

A:

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: k + 6 k+6 Hint: 126 = -te 3 Assume that n is a positive integer. Compute k+1 k3 + 3k² + 2k* k=1

A:

Q: Part 1. Obtain a solution to the following differential equation valid near the origin: 1.] For the…

A: Part 1. Given equation is y''+3xy'+7y=0   4. To Find: The recurrence relation for n≥2   5. To Find:…

Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence.  It was invented by German mathematician Carl Friedrich Gauss.