Consider the geometric sequence Po = 3, P₁ = 9, P₂ = 27, ... a) Find the common ratio R = b) Use the geometric sum formula to find the sum Po + P₁ + ... + P10.
Consider the geometric sequence Po = 3, P₁ = 9, P₂ = 27, ... a) Find the common ratio R = b) Use the geometric sum formula to find the sum Po + P₁ + ... + P10.
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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![**Geometric Sequence and Sum Calculation**
Consider the geometric sequence \( P_0 = 3 \), \( P_1 = 9 \), \( P_2 = 27 \), \( \ldots \)
**a) Finding the Common Ratio **
To find the common ratio (R), observe the progression of the sequence:
\( P_1 = 9 \)
\( P_0 = 3 \)
Common ratio \( R = \frac{P_1}{P_0} = \frac{9}{3} = 3 \)
Similarly,
\( P_2 = 27 \)
\( P_1 = 9 \)
Common ratio \( R = \frac{P_2}{P_1} = \frac{27}{9} = 3 \)
Hence, the common ratio \( R \) is:
\[ R = \boxed{3} \]
**b) Use the geometric sum formula to find the sum \( P_0 + P_1 + \ldots + P_{10} \)**
The sum \( S_n \) of the first \( n+1 \) terms of a geometric sequence is given by the formula:
\[ S_n = P_0 \frac{1 - R^{n+1}}{1 - R} \]
For \( n = 10 \):
\[ S_{10} = 3 \frac{1 - 3^{10+1}}{1 - 3} \]
\[ S_{10} = 3 \frac{1 - 3^{11}}{1 - 3} \]
\[ S_{10} = 3 \frac{1 - 177147}{-2} \]
\[ S_{10} = 3 \frac{-177146}{-2} \]
\[ S_{10} = 3 \times 88573 \]
\[ S_{10} = 265719 \]
So, the sum \( P_0 + P_1 + \ldots + P_{10} \) is:
\[ \boxed{265719} \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1c391162-9195-45f3-90d3-bf22c5f933af%2F9a38e323-5b5c-41fb-a576-cc5197c9310d%2Fw4ieei_processed.png&w=3840&q=75)
Transcribed Image Text:**Geometric Sequence and Sum Calculation**
Consider the geometric sequence \( P_0 = 3 \), \( P_1 = 9 \), \( P_2 = 27 \), \( \ldots \)
**a) Finding the Common Ratio **
To find the common ratio (R), observe the progression of the sequence:
\( P_1 = 9 \)
\( P_0 = 3 \)
Common ratio \( R = \frac{P_1}{P_0} = \frac{9}{3} = 3 \)
Similarly,
\( P_2 = 27 \)
\( P_1 = 9 \)
Common ratio \( R = \frac{P_2}{P_1} = \frac{27}{9} = 3 \)
Hence, the common ratio \( R \) is:
\[ R = \boxed{3} \]
**b) Use the geometric sum formula to find the sum \( P_0 + P_1 + \ldots + P_{10} \)**
The sum \( S_n \) of the first \( n+1 \) terms of a geometric sequence is given by the formula:
\[ S_n = P_0 \frac{1 - R^{n+1}}{1 - R} \]
For \( n = 10 \):
\[ S_{10} = 3 \frac{1 - 3^{10+1}}{1 - 3} \]
\[ S_{10} = 3 \frac{1 - 3^{11}}{1 - 3} \]
\[ S_{10} = 3 \frac{1 - 177147}{-2} \]
\[ S_{10} = 3 \frac{-177146}{-2} \]
\[ S_{10} = 3 \times 88573 \]
\[ S_{10} = 265719 \]
So, the sum \( P_0 + P_1 + \ldots + P_{10} \) is:
\[ \boxed{265719} \]
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