About Geometric Sequences and Series:
A geometric sequence, which is also known as a geometric progression is a sequence in which each term after the first is obtained by multiplying the preceding term by a fixed nonzero real number. The fixed nonzero real number is known as the common ratio.
Test Objectives
- Demonstrate the ability to find the common ratio
- Demonstrate the ability to find the nth term of a geometric sequence
- Demonstrate the ability to evaluate a geometric series
#1:
Instructions: Find the common ratio and the explicit formula.
$$a)\hspace{.2em}1, 4, 16, 64,...$$
Instructions: Find the term named and the explicit formula.
$$b)\hspace{.2em}2, -6, 18, -54,...$$ $$\text{Find:} \, a_{11}, a_{n}$$
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#2:
Instructions: Find the term named and the explicit formula.
$$a)\hspace{.2em}a_{4}=27, a_{1}=1$$ $$\text{Find:} \, a_{12}, a_{n}$$
$$b)\hspace{.2em}a_{1}=4, a_{6}=-972$$ $$\text{Find:} \, a_{11}, a_{n}$$
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#3:
Instructions: Find the term named and the explicit formula.
$$a)\hspace{.2em}a_{4}=-16, a_{5}=32$$ $$\text{Find:} \, a_{10}, a_{n}$$
$$b)\hspace{.2em}a_{3}=-48, a_{5}=-768, a_{10} < 0$$ $$\text{Find:} \, a_{8}, a_{n}$$
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#4:
Instructions: Evaluate each geometric series.
$$a)\hspace{.2em}\sum_{k=1}^{8}-3 \cdot 2^{k - 1}$$
$$b)\hspace{.2em}\sum_{j=1}^{10}3 \cdot 2^{j}$$
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#5:
Instructions: Evaluate each geometric series.
$$a)\hspace{.2em}\sum_{i=1}^{6}4 \cdot 3^{i + 1}$$
$$b)\hspace{.2em}\sum_{m=1}^{\infty}-2 \cdot \left(\frac{1}{2}\right)^{m - 1}$$
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Written Solutions:
#1:
Solutions:
$$a)\hspace{.2em}r=4$$ $$a_{n}=4^{n - 1}$$
$$b)\hspace{.2em}a_{11}=118{,}098$$ $$a_{n}=2 \cdot (-3)^{n - 1}$$
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#2:
Solutions:
$$a)\hspace{.2em}a_{12}=177{,}147$$ $$a_{n}=3^{n - 1}$$
$$b)\hspace{.2em}a_{11}=236{,}196$$ $$a_{n}=4 \cdot (-3)^{n - 1}$$
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#3:
Solutions:
$$a)\hspace{.2em}a_{10}=-1024$$ $$a_{n}=2 \cdot (-2)^{n - 1}$$
$$b)\hspace{.2em}a_{8}=-49{,}152$$ $$a_{n}=-3 \cdot 4^{n - 1}$$
Sign pattern from n = 3 to n = 10; given: a10 < 0
| n-value | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|---|---|---|---|---|---|---|---|---|
| r = +4 | − | − | − | − | − | − | − | − |
| r = −4 | − | + | − | + | − | + | − | + |
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#4:
Solutions:
$$a)\hspace{.2em}{-}765$$
$$b)\hspace{.2em}6138$$
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#5:
Solutions:
$$a)\hspace{.2em}13{,}104$$
$$b)\hspace{.2em}{-}4$$
