4. For each of the following expressions, write down an equivalent expression where the last term in the sum is outside the summation. (a) 17 E 2*. k=0 (b) 21 E (3k? – 2k + 1). k=-2 E (3k? – 2k + 1). k=-2

Q: For each of the summations given below, use the formula for the sum of the first n integers either…

A: Please refer the attached image for complete solution.

Q: Q3. Find the sum of the following series 1 1 n=3 'n²-4 зп-1.

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Q: Σ The value of the sum: + 2n is k=1

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Q: Find the value of the summation. 4 E 2* = k=1

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Q: Write the following expression as a single summation. ΤΗΣΗΣ 7· Σ (k2 + 8) + 9· Σ (2 - 1) = Σ k = 1 k…

A: Note :- These are multiple questions and according to our comapny guideline we can do maximum 1…

Q: 1. Write the following summation as a single summation: 3 (2k – 3) +(4- 5k) k=1 k=1

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Q: The sum from k=1 to infinity of: (k!/k3)

A: Given series is:

Q: Express the sum 3 + 5 + 7+ ... + (2k-1) using E-notation.

A: The given sum is 3+5+7+…+2k-1. Express the sum 3+5+7+…+2k-1 using ∑ notation as follows.

Q: Simplify by combining the two sums and find an erplicit formula. That is, your answer should not…

A: Simplify by combining the two sums and find an explicit formula. That is, your answer should not…

Q: Use the ratio test for the summation of (2n)!/n^9 when n equals 1 to infinity

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Q: b) Find the sum of the following formula. Please show your work 100 E(k² + 3k+2 +1) k=1

A: A sequence is a set of numbers that has a particular order or obeys a general form. The nth term of…

Q: 19. Find the sum using the summation formulas. 105 > (5k? – 7k + 14) k=1

A:

Q: For each of the following expressions, write down an equivalent expression where the last term in…

A: (a) The given expression is: ∑j=-218 2j Notice that the runs from -2 to 18. The last term is in the…

Q: Σ (²-2) for j=1 to 50. The summation equals

A: NOTE: Refresh your page if you can't see any equations. .

Q: What is the summation from n = 1 to infinity, of  (3n+1) / (5n + 4) ?

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Q: Simplify by comnbining the two sums and find an explicit formula. That is, your answer should not…

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Q: E-)k + 2) 9.

A: Solution is given below:

Q: Express the following sum in closed form. (2+3k)² = k 1 Hint: Start by multiplying out (2 + 3k)?.…

A: To write sum in closed form

Q: Q1. Express the sequence: 12 2 x(n) 3 0 As summed and shifted unit steps. n = 0 n = 1 n = 2…

A: Sol

Q: Sn denotes the sum of the first n terms of an AP prove that S30 = 3 (S20-S10

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Q: Write the following expression as a single summation. S (7k? + 4) + 2· (6k² – 1) = S %3D k = 1 k = 1…

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Q: What is the formula for the following summation? n=1 c(n) Option 1 Option 2 n(n+1) n(n+1)(2n+ 1) 21…

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Q: Find the value of the summation. 3 %3D k=1

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Q: Consider the series Σ (x – 8)4 5- 2! (x – 8)6 25 · 3! (x – 8)8 125 · 4! an = (x – 8)² + +... n=no…

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Q: Find the sum. 6 E k2 k = 1

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Q: Express the following sum in closed form. Σ 2 k > (4 +3 in k=1 -(+-2 ). k Hint: Start by multiplying…

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Q: 1. Find the value of the following summations : 3 2 5 7 b. 2 3 n= 0 а. с. n= 1 n = 4

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Q: Write the following expression as a single summation in terms of k. k k+ 2 k=1 m +3

A: The given series is ∑k=1mkk+2+m+1m+3. Rewrite the above series as follows.

Q: What is the summation from n = 1 to infinity, of  (3^(n+1)) / 5^n + 1 ?

A: Given that,       ∑n=1∞3n+15n+1

Q: Find the exact sum of the following series. 8∞ 1 -1 (4k-3) (4k+1)

A: A series is , a description of the operation of adding infinitely many quantities, one after the…

Q: Q2-)Determine the sum of the following series: 4 1- 2n=1 (4n-3)(4n+1). 2- 2n=0

A: 1. We use partial fractions to find the sum of the given series. 2. We use the sum of geometric…

Q: what is the summation where k = 1 of e^-2k

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Q: Find the exact sum of the following series. 1+ln(3)+(ln(3))2 /2+(ln(3))3 /6+(ln(3))4 /24+...

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Q: Which of the following is equal to this sum: n n 2Σ 3k2- k) - 4 Σ (-k+ 1) k = 1 k = 1

A: We use the following results to find the given sum:

Q: Find the sum S given by S=1² +2²x+3x²+4² x³ +0 for x < 1

A: A good problem. We shall look at the famous geometric series \sum x^n and obtain S by repeated…

Q: Suppose young Gauss had been asked to compute the sum 6 + 12 + 18 + 24 + ... + 1,494 + 1,500 1. How…

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Q: Find the value of the summation. (k2 – 3) = 46 o 2 k= 0

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Q: Compute the summation. 5 E (k + 3) = k = 1

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Q: Ex: when a sequence of n integers is added, The first being I and the last being n, the sum is Sum =…

A: Given: The sum of first npositive integers is given by, sum=n22+n2 To find: For n=100, 1000, 10000,…

Q: 3 k=1

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Q: Expand the following summation notation and vice versa. X12+ X13+ X4+ x;5 1. 2 2. Σ(Χ-Y) |

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Q: Which of the following is equal to this sum: 2Σ 8k2- k) -4 Σ (-k+1) k = 1 k 1 (6K2 - 2k + 4) a. k =…

A: Given : 2∑k=1n(3k2−k)−4∑k=1n(−k+4)

Q: 13 E(k- 1) – 8(k – 1) – 13] k=3

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Q: Write the following expression as a single summation in terms of k. m k Σ m + 1 Σ %3D k + 5 k = 1 m…

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Q: ΣΣ8.5 = t=1j=4

A: As per our guidelines we are supposed to answer only one question. Here first question answered.

Q: The sum of the series 2 3n-1 - 1 is given by n=1 6-1

A: Given series is an=3n-1-16n-1

Q: Write the following expression as a single summation in terms of k. m + 1 V k m + 1 k + 8 m + 9 k =…

A: To write the given expression as a single summation in terms of K.

Continuous Probability Distributions

Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.

Normal Distribution

Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!