Explanation Approach: 1) An arithmetic sequence is a sequence in which each term after the first term differs from the preceding term by a fixed constant amount called the common difference of the sequence. 2) The nth term of an arithmetic sequence with first term a1 and common difference d is given by a n = a 1 + ( n − 1) d . 3) The sum of the first n terms of an arithmetic sequence is given by n ( a 1 + a n ) 2 . Calculation: Consider the following sequence, 1, 1.5, 2.0, 2.5,... Here, d = second term – first term = 1.5 – 1 = 0.5 Use the formula of n th term of an arithmetic sequence to find a 20

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ISBN: 9780134434681

Author: Tom Pirnot

Publisher: PEARSON

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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.

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Chapter 6.6, Problem 17E

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The specified term a_n and then find the sum of the terms from a₁ to a_n, inclusive for the given arithmetic sequence.

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Chapter 6.6, Problem 16E

Chapter 6.6, Problem 18E

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(b) An otherwise fair six-sided die has been tampered with in an attempt to cheat at a dice game. The effect is that the 1 and 6 faces have a different probability of occurring than the 2, 3, 4 and 5 faces. Let θ be the probability of obtaining a 1 on this biased die. Then the outcomes of rolling the biased die have the following probability mass function. Table 1 The p.m.f. of outcomes of rolls of a biased die Outcome 1 2 3 4 5 6 Probability θ 1 4 (1 − 2θ) 1 4 (1 − 2θ) 1 4 (1 − 2θ) 1 4 (1 − 2θ) θ (i) By consideration of the p.m.f. in Table 1, explain why it is necessary for θ to be such that 0 < θ < 1/2. [2] (ii) The value of θ is unknown. Data from which to estimate the value of θ were obtained by rolling the biased die 1000 times. The result of this experiment is shown in Table 2. Table 2 Outcomes of 1000 independent rolls of a biased die Outcome 1 2 3 4 5 6 Frequency 205 154 141 165 145 190 Show that the likelihood of θ based on these data is L(θ) = C θ395 (1 − 2θ) 605 ,…

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The specified term a n and then find the sum of the terms from a 1 to a n , inclusive for the given arithmetic sequence.

Solution Summary: The author explains how an arithmetic sequence differs from the preceding term by the fixed constant amount called the common difference of the sequence.

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To find:

The specified term a_n and then find the sum of the terms from a₁ to a_n, inclusive for the given arithmetic sequence.

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Chapter 6 Solutions

The specified term a n and then find the sum of the terms from a 1 to a n , inclusive for the given arithmetic sequence.

Solution Summary: The author explains how an arithmetic sequence differs from the preceding term by the fixed constant amount called the common difference of the sequence.

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To find: The specified term an and then find the sum of the terms from a1 to an, inclusive for the given arithmetic sequence.

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Chapter 6 Solutions

To find:

The specified term a_n and then find the sum of the terms from a₁ to a_n, inclusive for the given arithmetic sequence.