### 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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Description: ### 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 te

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Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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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\

Expert Solution

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12)

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

It is given that in an arithmetic sequence, $a_{5} = - 46 a n d a_{10} = - 91$ .

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

$a_{5} = a + 4 d = - 46 (1) a_{10} = a + 9 d = - 91 (2)$

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