Compute o, (1000), o,(750), and o, (1024). Hints: the divisor-sum function or(n). When k is any number and n is a positive integer, let or (n) denote the sum of the k" powers of the positive divisors of n. For example: o (10) = 1° + 2° + 5° + 10° = 1+1+1+1 = 4 0 (10) = 1' + 21 + 51 + 101 0, (10) = 12 + 22 + 52 + 102 =1+4+25 + 100 = 130 18 1+2+5+10

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Q: n E(-1)* k + k)"-1, n2 1. k=0

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Q: 13 e, thank you.

A: e)Given that, sum of the first n terms of the given series is

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Q: n! diverges because an > 0. for all n > 1. True False

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Q: Prove by induction. 3F, = Fn+2+ Fn-2 for n > 2

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Q: 2. Let an = 3n + 2 for n = 0, 1, 2, ... (i ) Find ao, a1 and a2. (ii ) Show that a2 = 4a1 – 3ao.…

A: Answer and explanation is given below..

Q: (b) Show that k-1 (k+1) n n+1 for every n ≥ 1.

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Q: Solve the following recursion relation T(n) = n* T(n-1) if n>1 = 1.…

A: Explanation of the answer is as follows

Q: (b) Use induction and the Product Rule to prove d. -I" = nr"-1 for all n E No.

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Q: Find a number n e N such that 2" > 102000 and 2" has among its 2000 last digits at least 1000…

A: We will use logarithm to solve the problem

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Q: Prove that then the sequence {"(1+i) belongs to l1. n3"

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Q: b) Prove that all numbers n > 20 can be represented as n = 4 * i +7*j for some non- negative i and j…

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Q: Find the a0, a1, a2, and lim an for the following sequences: 3 x 10 = 30. an = (7+3n)/(5-2n) an =…

A: We can solve this by using definition of sequences

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A: Let's write some initial terms of the sequence and observe the pattern.

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A: Given: an+1=n2.ana1=3a3-10a2=?

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A: Given  y = ∑n=0∞ cnxn ∴ y' = ∑n=1∞ cn n xn-1 ∴ y'' = ∑n=2∞ cn n (n-1) xn-2

Q: Q3/A/ Prove that (") + („",) = (*). n+: \k+1. n %3| k+1

A: Given:

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Q: Find a generating function for the sequence (ao, a1,...) where a, is the number of nonnegative…

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Q: Let n = p¹, be a factorization of n into distinct primes p₁,..., Pk. Then, i=1 p(n) = n(1 - 1) (1 -…

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Q: A sequence {am} of real numbers is defined recursively by a1 = 1, a2 = 2 and a, = (i–1)a; for n > 3.…

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Q: Find the recurrence relation of the sequence defined by T(1) = 2 T(n) = 2n * T(n-1), n &gt; 1

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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 =…

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Q: [4.0091, 0.12319, 1.2341]" and y = [-1.1021, .35449, 3.5449]". Using 3 Let x = decimal-digit…

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Q: 1. Prove by induction that if y = , then d"y = (-1)"n!x¯n-1, dx" where n! = 1 · 2..... n is the…

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Q: 3. The Fibonacci sequence is defined recursively by to = 0, t1 = 1, t = tk-1+tk-2 for t > 1. Use…

A: The given problem is to prove the following Fibonacci sequence using induction method.

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Q: Show that the sum of the first n positive odd integers,1 + 3 + 5 + g+ (2n - 1),is n2.

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Q: (4) Determine the generating function for the sequence ho, h1, ..., where hn = #((a1, a2, a3, a4) |…

A: By definition of Generating Function,                The Generating function for the sequence…

Q: Let f(n) = 5" – 4. (a) Evaluate f when n is 1, 2, 3, 4, and 5. (b) For what positive integers n is…

A: The function is  :

Q: show that the sum of this: pi/(3sqrt(3))=(∑∞ n=0 ((-1)n (1/2)3n+1))/(3n+1) + ∑∞ n=0  ((-1)n…

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Q: For any integer n2 1. prove that: (a) 1+2+3+..+ (n – 1) + n + (n – 1) + . .. +3+2+ 1 = n². (b) +...+…

A: Note: According to Bartleby guidelines; for more than three questions asked, only the first three…

Unitary Method

The word “unitary” comes from the word “unit”, which means a single and complete entity. In this method, we find the value of a unit product from the given number of products, and then we solve for the other number of products. 

Speed, Time, and Distance

Imagine you and 3 of your friends are planning to go to the playground at 6 in the evening. Your house is one mile away from the playground and one of your friends named Jim must start at 5 pm to reach the playground by walk. The other two friends are 3 miles away.

Profit and Loss

The amount earned or lost on the sale of one or more items is referred to as the profit or loss on that item.

Units and Measurements

Measurements and comparisons are the foundation of science and engineering. We, therefore, need rules that tell us how things are measured and compared. For these measurements and comparisons, we perform certain experiments, and we will need the experiments to set up the devices.