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What s the difference between a sequence (also called a progression) and a series
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Answer 19-1
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A sequence is a list of numbers or variables Such a list can contain anywhere from two to infinitely many elements A series is the sum of the elements in a specific sequence
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What s an arithmetic sequence
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An arithmetic sequence is a list of numbers that starts at a certain value, and then increases or decreases at a constant rate after that Therefore, each element is larger or smaller than its predecessor by a certain fixed amount
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What s the general form of a finite arithmetic sequence of real numbers What s the general form of an infinite arithmetic sequence of real numbers
Answer 19-3
The general form of a finite arithmetic sequence Sfin is Sfin = s0, (s0 + c), (s0 + 2c), (s0 + 3c), , (s0 + nc) where s0 is a real number representing the first element, c is a real number representing the sequence constant, and n is a positive integer In this case, the sequence has n + 1 elements The general form of an infinite arithmetic sequence Sinf is Sinf = s0, (s0 + c), (s0 + 2c), (s0 + 3c), where s0 is a real number representing the first element, and c is a real number representing the sequence constant The ellipsis () tells us that the sequence continues without end
Question 19-4
What are the partial sums of an infinite arithmetic sequence
Answer 19-4
When we add up the elements of a numeric sequence, we get another list of numbers called the sequence of partial sums For Sinf described in Answer 19-3, the first five partial sums are s0 s0 + s0 + c s0 + s0 + c + s0 + 2c s0 + s0 + c + s0 + 2c + s0 + 3c s0 + s0 + c + s0 + 2c + s0 + 3c + s0 + 4c which can be simplified to s0 2s0 + c 3s0 + 3c
Part Two 433
4s0 + 6c 5s0 + 10c
Question 19-5
What s a geometric sequence
Answer 19-5
A geometric sequence is a list of numbers with a starting value that s repeatedly multiplied by a constant factor If we take any element in the sequence (except the first one) and divide it by its predecessor, we always get the same constant
Question 19-6
What s the general form of a finite geometric sequence of real numbers What s the general form of an infinite geometric sequence of real numbers
Answer 19-6
The general form of a finite geometric sequence Tfin is Tfin = t0, t0k, t0k2, t0k3, t0k4, , t0kn where t0 is a real number representing the first element, k is a real number representing the sequence constant, and n is a positive integer In this case, the sequence has n + 1 elements The general form of an infinite geometric sequence Tinf is Tinf = t0, t0k, t0k2, t0k3, t0k4, where t0 is a real number representing the first element, and k is a real number representing the sequence constant
Question 19-7
What are the partial sums of an infinite geometric sequence
Answer 19-7
For Tinf described in Answer 19-6, the first five partial sums are t0 t0 + t0k t0 + t0k + t0k2 t0 + t0k + t0k2 + t0k3 t0 + t0k + t0k2 + t0k3 + t0k4
Review Questions and Answers
which can be simplified to t0 t0 (1 + k) t0 (1 + k + k2) t0 (1 + k + k2 + k3) t0 (1 + k + k2 + k3 + k4)
Question 19-8
How can summation notation be used to symbolize a finite series How can summation notation be used to symbolize an infinite series
Answer 19-8
Suppose we have a series with n terms a1 + a2 + a3 + + an 2 + an 1 + an We can symbolize it by writing
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