Answered: 1 1 S₁) = 1 + = + = =++ (~EN). S(n)… | bartleby

Algebra & Trigonometry with Analytic Geometry

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.5: The Binomial Theorem

Problem 16E

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1
If S(n) = 1 + = + = +
23
+
S(1) +
+-(ne N), then prove that
n
S(2) + ... + S(n-1) = (nS(n) − n) or (nS(n-1) − n + 1)
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2. Suppose that in Example 2.27, 400 units of food A, 500 units of B, and 600 units of C are placed in the test tube each day and the data on daily food consumption by the bacteria (in units per day) are as shown in Table 2.7. How many bacteria of each strain can coexist in the test tube and consume all of the food? Table 2.7 Bacteria Strain I Bacteria Strain II Bacteria Strain III Food A 1 2 0 Food B 2 1 3 Food C 1 1 1
We now need to attempt to find an exact value for n such that P(r ≥ 1) = 0.99. Clearly, if the bank has any chance of detecting a burglar they will need to have at least one alarm installed. So, we know that, minimally, n must be at least 1.Since n must be at least 1, suppose we start by trying n = 2, where there will be two alarms. When there are two alarms, r ≥ 1 means either one or both alarms could detect the burglar. If neither alarm detects the burglar it is considered a failure. Therefore, P(r ≥ 1) can be calculated using either of the following formulas. P(r ≥ 1) = P(1) + P(2) P(r ≥ 1) = 1 − P ( ___ ) fill in the blank
If s(n)=3n2−3n+5s(n)=3n2−3n+5, then s(n)=2s(n−1)−s(n−2)+cs(n)=2s(n−1)−s(n−2)+c for all integers n≥2n≥2. What is the value of c ?
Consider the numbers sn defined by sn = 1 + 3 + 5 + 7 + … + (2 * n + 1), for all n ∈ ℕ. That is, sn is the sum of the first n + 1 odd numbers.
Show that Sn is solvable when n ≤ 4.
Let n ∈ Z, Is it true 4 | n4 – n? Prove it or counterexample.
FOR N=195, by fermat's theorem an-2 =
if we had some f(z)= an infinite sum of z^n how would we solve this?
4. The lifetime, in years, of a type of small electric motor operating under adverse conditions is T ~Γ(6, 3.6). Whenever the motor fails, it is replaced with another of the same type. Find theprobability that fewer than 6 motors fail within one year.
Let q1, q2, q3, . . . be defined by q1 = 2, qn = 3qn−1 − 1 for n ≥ 2. Show byinduction that for all n ≥ 1, qn =1/2(3n + 1).
Prove for r ≠ 1 that ∑_(k=0)^(n-1)(rk)=(1-rn)/(1-r). If ∑_(k=0)^(n-1)(rk), then compute s(1−r) = s−rs, and solve for s.
6. Below is an incomplete proof Prove that 3n2 > 2(n+1)2 for n ≥ 5 3n2 > 2(n+1)2 Is equivalent to 3n2 > 2(n2 + 2n +1) Is equivalent to 3n2 > 2n2 + 4n +2 Is equivalent to n2 > 4n +2 Finish the proof, by showing why n2 > 4n +2 Hint: you’ll need to use the fact that n ≥ 5.

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