Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 17: Problem 383
The first term of a geometric progression is 27, the nth term is \(32 / 9\), and the sum of n terms is \(665 / 9\). Find \(\mathrm{n}\) and \(\mathrm{r}\).
Short Answer
Expert verified
The values of the variables are: \(n = 6\) and \(r = \frac{1}{3}\).
Step by step solution
01
Find the formula for the nth term of a geometric progression
A geometric progression (GP) is a sequence of numbers where each term is found by multiplying the previous term by a constant, called the common ratio (r). The formula to find the nth term (T_n) of a GP with the first term (T_1) is:
T_n = T_1 * r^(n-1)
In our case, T_1 = 27, and T_n = 32/9.
02
Find the formula for the sum of the first n terms of a geometric progression
The formula for the sum of the first n terms (S_n) of a GP with the first term (T_1) and common ratio (r) is:
S_n = T_1 * (1 - r^n) / (1 - r)
In our case, S_n = 665/9.
03
Write the given information in terms of formulas
We have the information about T_1, T_n and S_n:
T_1 = 27
T_n = 32 / 9
S_n = 665 / 9
Now we use the formulas that we've written down in the first two steps:
T_n = T_1 * r^(n-1) -> That is, 32/9 = 27 * r^(n-1)
S_n = T_1 * (1 - r^n) / (1 - r) -> That is, 665/9 = 27 * (1 - r^n) / (1 - r)
04
Solve the equations to find r and n
First, let's isolate r^(n-1) in the equation for T_n:
r^(n-1) = (32/9) / 27 = (32/9) * (1/27) = 32/243
Now substitute r^(n-1) in the equation for S_n:
665/9 = 27 * (1 - r^n) / (1 - r)
Using the fact that r^(n-1) = 32/243, we have:
r^n = r * r^(n-1) = r * (32/243)
Now substitute r^n into the equation for S_n:
665/9 = 27 * (1 - r * (32/243)) / (1 - r)
Now, we can solve for r:
665 = 243 * (1 - 32r) / (9 - 9r)
Multiplying both sides by (9 - 9r) will result in:
665 * (9 - 9r) = 243 * (1 - 32r)
Dividing both sides by 9 gives:
665 * (1 - r) = 81 * (1 - 32r)
Expanding both sides:
665 - 665r = 81 - 2592r
Finally, solving for r:
r = (81-665) / (2592-665) = 1/3
Now we will use the relation r^(n-1) = 32/243 to find the value of n:
(1/3)^(n-1) = 32/243
Taking the logarithm base (1/3) on both sides:
n-1 = log_{1/3}(32/243)
n-1 = 5
n = 6
So, the values of the variables are: n = 6 and r = 1/3.
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
nth term formula
The nth term formula is essential for understanding a geometric progression. This formula allows you to find any term in the sequence without needing to know all the previous terms. The general form of this formula is given as:\[ T_n = T_1 \cdot r^{n-1} \]where:
- \( T_n \) is the nth term you want to find,
- \( T_1 \) is the first term of the sequence,
- \( r \) is the common ratio,
- \( n \) is the term number.
sum of n terms formula
The sum of the first n terms in a geometric progression can be calculated using this important formula:\[ S_n = T_1 \cdot \frac{1 - r^n}{1 - r} \]where:
- \( S_n \) is the sum of the first n terms,
- \( T_1 \) is again the first term,
- \( r \) is the common ratio,
- \( n \) is the number of terms to sum.
common ratio
The common ratio \( r \) is a pivotal value in a geometric progression. It indicates how each term in the sequence is derived from the previous one by multiplication. The value of \( r \) can significantly affect the progression's growth or decay.To find the common ratio, you typically rearrange the nth term formula as discussed earlier. In the exercise, we derived that:\[ r^{n-1} = \frac{32}{243} \]By manipulating both the nth term and the sum formula, you can isolate \( r \) and subsequently solve it using known methods such as logarithms.In our example, solving the equations gave us a common ratio of \( \frac{1}{3} \). This indicates that each term is one-third of the previous one, showing a decreasing sequence. Understanding \( r \) helps you predict how the sequence behaves over its course.
sequence
A geometric sequence, or progression, is a series of numbers where each term is calculated by multiplying the previous term by a constant called the common ratio. Recognizing patterns in sequences helps in deducing the rules and formulas.In our case, the sequence starts with the first term \( T_1 = 27 \) and continues by multiplying by the common ratio \( r \). Knowing both the nth term and the sum formula, you can solve for missing variables like \( n \) and \( r \). In the given exercise, the sequence could look like: 27, 9, 3, 1, etc., showing values diminishing by a ratio of \( \frac{1}{3} \).Better understanding sequences not only aids in solving equations but also in grasping the broader mathematical principles governing how numbers progress. Studying these series is both a practical tool and a way to recognize the elegance and order within mathematics.
