Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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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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### Arithmetic and Geometric Sequences and Series **12. Find \(a_1\) of an arithmetic sequence when \(a_5 = -46\) and \(a_{10} = -91\).** Options: - a) \(-\frac{6}{5}\) - b) \(-10\) - c) \(-\frac{5}{6}\) - d) \(16\) **13. Find the sum of the first 20 terms of an arithmetic series with \(a_3 = 13\) and \(a_{12} = 58\).** Options: - a) \(1010\) - b) \(1020\) - c) \(950\) - d) \(1050\) --- ### Arithmetic Sequence Formula **14. Find the formula for the \(n\)th term of the arithmetic sequence \(-8, -5, -2, \ldots\).** Options: - a) \(a_n = -8 + 3n\) - b) \(a_n = 3n - 11\) - c) \(a_n = 3n - 5\) - d) \(a_n = n + 3\) --- ### Series Evaluation **15. Compute the exact sum \( \frac{1}{2} + \frac{1}{2^2} + \frac{1}{2^3} + \cdots + \frac{1}{2^{10}} \).** Options: - a) \(1 + \frac{1}{2^{10}}\) - b) \(\frac{1}{2^{10}}\) - c) \(1 - \frac{1}{2^{10}}\) - d) \(\frac{1}{2^{10}} - 1\) --- ### Geometric Sequence Formula **16. Find the formula for the general term \(a_n\) of a geometric series with \(a_3 = -\frac{1}{8}\) and \(a_7 = -\frac{1}{128}\).** Options: - a) \(a_n = \frac{1}{2}(-\frac{1}{2})^{n-1}\) - b) \(a_n = (\frac{1}{2})^n\
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Transcribed Image Text:### Arithmetic and Geometric Sequences and Series **12. Find \(a_1\) of an arithmetic sequence when \(a_5 = -46\) and \(a_{10} = -91\).** Options: - a) \(-\frac{6}{5}\) - b) \(-10\) - c) \(-\frac{5}{6}\) - d) \(16\) **13. Find the sum of the first 20 terms of an arithmetic series with \(a_3 = 13\) and \(a_{12} = 58\).** Options: - a) \(1010\) - b) \(1020\) - c) \(950\) - d) \(1050\) --- ### Arithmetic Sequence Formula **14. Find the formula for the \(n\)th term of the arithmetic sequence \(-8, -5, -2, \ldots\).** Options: - a) \(a_n = -8 + 3n\) - b) \(a_n = 3n - 11\) - c) \(a_n = 3n - 5\) - d) \(a_n = n + 3\) --- ### Series Evaluation **15. Compute the exact sum \( \frac{1}{2} + \frac{1}{2^2} + \frac{1}{2^3} + \cdots + \frac{1}{2^{10}} \).** Options: - a) \(1 + \frac{1}{2^{10}}\) - b) \(\frac{1}{2^{10}}\) - c) \(1 - \frac{1}{2^{10}}\) - d) \(\frac{1}{2^{10}} - 1\) --- ### Geometric Sequence Formula **16. Find the formula for the general term \(a_n\) of a geometric series with \(a_3 = -\frac{1}{8}\) and \(a_7 = -\frac{1}{128}\).** Options: - a) \(a_n = \frac{1}{2}(-\frac{1}{2})^{n-1}\) - b) \(a_n = (\frac{1}{2})^n\
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12)

It is known that the nth term of an arithmetic sequence is given by an=a+n-1d where a is the first term and d is the common difference.

It is given that in an arithmetic sequence, a5=-46  and  a10=-91.

By using the nth term formula, the 5th and 10th terms can be written as follows.

a5=a+4d=-46                                                            1a10=a+9d=-91                                                          2

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