Write the sum without sigma notation and evaluate it. 3 11n 17) E (-1)k sin 2 k = 1 117 A) -sin 11T + sin 11n sin 2 117 B) -sin 11n + sin 117 sin = -1 2 2 2 2 11n C) -sin 2 117 117 D) -sin 11n + sin 2 117 sin 2 sin = 2 2

Q: Use the properties of summation and Theorem 4.2 to evaluate the sum. 72 n (n - 1) Theorem 4.2: i=1…

A: We have to evaluate the sum.

Q: 1. (2i–1) = 1+3+5+7+9+..+(2n – 1) = n².

A:

Q: n-(n+1)·(2n+1) 12 + 22 + 32 +... +n² ||

A: The solution is given as

Q: If z1=1-i,z2=-2+4i then evaluate: modulus of (z1+z2+1)/(z1-z2+i)

A:

Q: Which of the sums below is equivalent to 4sin (36)cos (5ß)? Select the correct answer below: O 4cos…

A: 2 cos(A) sin(B) = sin(A+B) - sin(A-B)

Q: Consider the sum (3)² + (8)² + (13)2 + · . + (15n – 2)2. | m 1 1 m(т + 1)] and k2 = (m(m + 1)(2m +…

A: Consider S=32+82+132+ . . . .+15n-22

Q: 5. Show by mathematical induction that the geometric sum can also be written by Zar' = a+ ar + ar…

A:

Q: Prove that 2n+1 > (n + 2) · sin(n) for all positive integers n.

A:

Q: Use induction to prove: for any integer n 2 0, 2. 3 = 3"+1 - 1. %3D j=0 Base case 2.3 = j= n = ,…

A: for any integer n≥0,∑j=0n2. 3j=3n+1-1

Q: Define a sequence of real numbers as follows: Let x1 = 1, and then for ever n ∈ ℕ, define xn+1 = (xn…

A: Given : a sequence of real numbers such that         x1=1 and then for every n∈ℕ, define xn+1=xn2+1…

Q: 6.60. Below is given a proof of a result. Which result is being proved and which proof technique is…

A:

Q: IM- Prove that n(n+ 1)(2n + 1)

A:

Q: Use sigma notation to write the sum. 7 [›( ² ) + ²] + [²( ² ) + ²] + ... + [>( ² ) + ²] X 7j 214 Σ j…

A: Given the sum is 714+2 + 724+2 + … + 784+2

Q: Consider the sum of odd integers: 1+3 +5+7+ 9 + ..+ (2n – 1) | Let the symbol Sk represent the sum…

A:

Q: Can i get help step by step with this problem?

A: The sigma notation used is as follows,

Q: Using induction, Prove that  2 + 6 + 10 + ... + (4n - 2)   = 2n2 for all n  ∊  N

A: According to the question, we need to prove that: 2 + 6 + 10 + ... + (4n - 2)   = 2n2 for all n  ∊…

Q: Assume C and D are chosen at random from {1, 2 . . . , n}. Let pn be the probability that C + D is a…

A: Given: C and D are chosen at random from {1, 2 . . . , n}. Then the number of pairs (c, d) with sum…

Q: fond the value of the double sum

A:

Q: (n+1)(5n+6) Prove that 3+Č(3+5j)= j=l

A:

Q: Can i get help step by step with this problem?

A: Consider the given series.

Q: Prove using mathematical induction that for all n E N. i E7-1(8i – 5) = 4n2 – n %3D ii. 52n - 1 is…

A:

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: Use direct proof to show that a. if m and n are odd integers their sum m+n is even. b. If n is odd…

A:

Q: Use mathematical induction to prove the following. 1. Prove that for all positive integer n,…

A:

Q: 2 13. Write the sum 2++ in sigma notation in 25 three ways: with a starting value of (a) k = 1, (b)…

A: a) The series can be written in compact form using sigma notation Here the terms are written…

Q: Solve for x. All final answers must be in Zn where n is the modulus. 6x mod 11 ≡1

A: Solving this is equivalent to finding multiplicative inverse of 6 in  Z11 . We have to find x such…

Q: 34. P(n): (cos 0 + i sin 0)" cos no + i sin nð

A: P(n): (cos θ+i sin θ)n=cos nθ+i sin nθ

Q: 126 126 Suppose that > a; = 19 and ) b; = 11. Compute the sum. %3D %3D i=1 i=1 126 E(- 5a; + 27b;) |…

A:

Q: Use mathematical induction to prove that 2n-1 (2i + 1) = 3n² i=n

A: Handwriting solution.

Q: Prove the following statement by mathematical induction. For every integer n ≥ 0, n + 1…

A: For every integer  n ≥ 0,  n + 1   summation i = 1 i · 2i = n · 2n + 2 + 2.

Q: Make a conjecture for each of the following Fibonacci sums, where Fn represents the nth Fibonacci…

A: Given, Fn represent the nth Fibonacci number,

Q: Let n ≥ 1 be an integer. Prove by induction 3 + 10 + 17 + · · · + 7n − 4 = n(7n − 1) / 2 Do not use…

A:

Q: 5 = t(n + 1) = t(n +2) = t(n + 3) holds for n = 3655 and 4503. (a) Verify that t (n) (b) When n =…

A:

Q: Intro to proofs Given : AB=DC PROVE x=8 2x+5 =AB 5x-19= CD

A:

Q: Write the sums without sigma notation. Then evaluatethem.

A: Sum of sequence: ∑k=13 (-1)k+1sinπk We need to put the k values in the above sequence as k=1,2,3…

Q: Prove by mathematical induction the summation 3n +7n (3i + 2) = 5+8+11+ 14+...+(3n +2) = (Write your…

A: Summation

Q: The modulus of 5 + 4i is * O-41 O-√41 O√41 041 The conjugate of (i-i²)³ is O-2-21 2+2i -2+2i Oi-1

A:

Q: Use direct proof to show that a. if m and n are odd integers their sum m+n is even. b. If n is odd…

A:

Q: Let Hn = the sum from i= 1 to n of 1/ifor any n ≥ 1. Prove or disprove that for any n ≥ 1H2n ≥n/4.

A: So, Hn looks like,             We will use mathematical induction to show that,…

Q: Expand and evaluate given the sum:

A: By the definition of summation, sum of n terms a1, a2,....., an is  written as ∑i=1nai i.e.…

Q: A sequence of numbers co, ¤1, X2, .. . is such that xo = 4 and, 6. %3D for each n > 1, 1. In 3n-1+2.…

A:

Q: Exercise 1. Show that there are infinitely many primes of the form 4n +3. Hint: you might find it…

A:

Q: n 2' = 2"+1 – 1 2n+1 i=0

A:

Q: Sequence is defined by 02 U。く2 n70: Un = V2- Un-1 Suppose: Po-Von and an= = Uzn+1 * Prove that the…

A: Given: Sequence is defined by 0<U0<2 n>0: Un=2-Un-1 Suppose : pn=U2n and  qn=U2n+1

Q: 2. 2i = 2+4 +6+8+10+...+ 2n = n(n+1). %3D i=1

A: We can prove this statement using the method of induction as below.

Q: Write out the sum. 2 (k+6) k= 1 Find the first, second, and last terms of the su in I (k + 6) =D•+O…

A:

Q: Write the sums without sigma notation. Then evaluatethem.

A: Given: ∑k=14-1kcoskπ. Writting the sums without sigma notation.…

Q: Use sigma notation to write the sum. + 3 + 2 + 3 + + 3 + j = 1

A:

Q: Use mathematical induction to prove the summation formula: 1 + 3 + 5 + . . . + (2n − 1) = n2

A: The solution is given as

Q: II. 1. Prove that for all integers n if 51n? then 5\n. 1 2. Prove : 1 1 1 n + Vn 21,nEZ + ... 1•2…

A: