Geometric Sequences and Series worksheet, answer key and bellwork - Free Printable

Problem Analysis:

The problem involves several parts related to geometric sequences and series. Let's break it down step by step and solve each part systematically.

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#### Part 1: Finding the First Four Terms of Each Geometric Sequence

We are given two geometric sequences:

1. Sequence (i): \( a = 3 \), \( r = -2 \)
2. Sequence (ii): \( a = 16 \), \( r = \frac{1}{4} \)

A geometric sequence is defined as:
\[ a, ar, ar^2, ar^3, \ldots \]

##### (i) For \( a = 3 \), \( r = -2 \):
- First term: \( a_1 = a = 3 \)
- Second term: \( a_2 = ar = 3(-2) = -6 \)
- Third term: \( a_3 = ar^2 = 3(-2)^2 = 3(4) = 12 \)
- Fourth term: \( a_4 = ar^3 = 3(-2)^3 = 3(-8) = -24 \)

Thus, the first four terms are:
\[ 3, -6, 12, -24 \]

##### (ii) For \( a = 16 \), \( r = \frac{1}{4} \):
- First term: \( a_1 = a = 16 \)
- Second term: \( a_2 = ar = 16 \left(\frac{1}{4}\right) = 4 \)
- Third term: \( a_3 = ar^2 = 16 \left(\frac{1}{4}\right)^2 = 16 \left(\frac{1}{16}\right) = 1 \)
- Fourth term: \( a_4 = ar^3 = 16 \left(\frac{1}{4}\right)^3 = 16 \left(\frac{1}{64}\right) = \frac{1}{4} \)

Thus, the first four terms are:
\[ 16, 4, 1, \frac{1}{4} \]

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#### Part 2: Finding the Indicated Term of Each Geometric Sequence

We are given three geometric sequences and asked to find specific terms.

##### (a) For \( a = 5 \), \( r = 3 \), find \( a_7 \):
The general term of a geometric sequence is given by:
\[ a_n = ar^{n-1} \]

Here, \( a = 5 \), \( r = 3 \), and \( n = 7 \):
\[ a_7 = 5 \cdot 3^{7-1} = 5 \cdot 3^6 \]
\[ 3^6 = 729 \]
\[ a_7 = 5 \cdot 729 = 3645 \]

##### (b) For \( a = 2 \), \( r = \sqrt{3} \), find \( a_6 \):
Here, \( a = 2 \), \( r = \sqrt{3} \), and \( n = 6 \):
\[ a_6 = 2 \cdot (\sqrt{3})^{6-1} = 2 \cdot (\sqrt{3})^5 \]
\[ (\sqrt{3})^5 = (\sqrt{3})^4 \cdot \sqrt{3} = (3^2) \cdot \sqrt{3} = 9\sqrt{3} \]
\[ a_6 = 2 \cdot 9\sqrt{3} = 18\sqrt{3} \]

##### (c) For \( a = 1 \), \( r = -\frac{1}{2} \), find \( a_{10} \):
Here, \( a = 1 \), \( r = -\frac{1}{2} \), and \( n = 10 \):
\[ a_{10} = 1 \cdot \left(-\frac{1}{2}\right)^{10-1} = \left(-\frac{1}{2}\right)^9 \]
\[ \left(-\frac{1}{2}\right)^9 = -\frac{1}{2^9} = -\frac{1}{512} \]
\[ a_{10} = -\frac{1}{512} \]

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#### Part 3: Using Formulas to Find Specific Terms

We are given two more geometric sequences and asked to find specific terms using formulas.

##### (i) For \( a = 2 \), \( r = \frac{1}{2} \), find \( a_6 \):
Using the general term formula:
\[ a_n = ar^{n-1} \]
Here, \( a = 2 \), \( r = \frac{1}{2} \), and \( n = 6 \):
\[ a_6 = 2 \cdot \left(\frac{1}{2}\right)^{6-1} = 2 \cdot \left(\frac{1}{2}\right)^5 \]
\[ \left(\frac{1}{2}\right)^5 = \frac{1}{32} \]
\[ a_6 = 2 \cdot \frac{1}{32} = \frac{2}{32} = \frac{1}{16} \]

##### (ii) For \( a = -3 \), \( r = -\frac{1}{3} \), find \( a_5 \):
Using the general term formula:
\[ a_n = ar^{n-1} \]
Here, \( a = -3 \), \( r = -\frac{1}{3} \), and \( n = 5 \):
\[ a_5 = -3 \cdot \left(-\frac{1}{3}\right)^{5-1} = -3 \cdot \left(-\frac{1}{3}\right)^4 \]
\[ \left(-\frac{1}{3}\right)^4 = \frac{1}{81} \]
\[ a_5 = -3 \cdot \frac{1}{81} = -\frac{3}{81} = -\frac{1}{27} \]

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#### Part 4: Population Growth Problem

We are given that the population of a city is 100,000 and it increases by 5% per year. We need to find the population after 10 years.

The population growth can be modeled as a geometric sequence where:
- Initial population \( P_0 = 100,000 \)
- Growth rate \( r = 5\% = 0.05 \)
- Population after \( n \) years is given by:
\[ P_n = P_0 \cdot (1 + r)^n \]

Here, \( n = 10 \):
\[ P_{10} = 100,000 \cdot (1 + 0.05)^{10} \]
\[ P_{10} = 100,000 \cdot (1.05)^{10} \]

Using a calculator to find \( (1.05)^{10} \):
\[ (1.05)^{10} \approx 1.628894626 \]

Thus:
\[ P_{10} = 100,000 \cdot 1.628894626 \approx 162,889.46 \]

Rounding to the nearest whole number:
\[ P_{10} \approx 162,889 \]

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Final Answers:

1. First four terms:
- (i) \( 3, -6, 12, -24 \)
- (ii) \( 16, 4, 1, \frac{1}{4} \)

2. Indicated terms:
- (a) \( a_7 = 3645 \)
- (b) \( a_6 = 18\sqrt{3} \)
- (c) \( a_{10} = -\frac{1}{512} \)

3. Using formulas:
- (i) \( a_6 = \frac{1}{16} \)
- (ii) \( a_5 = -\frac{1}{27} \)

4. Population after 10 years:
\[ \boxed{162,889} \]