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Magazine Chapter 9: Problem 80
In a geometric progression, if the ratio of the sum of first 5 terms to the sum of their reciprocals is 49 , and the sum of the first and the third term is 35 . Then the first term of this geometric progression is: [Online April 11, 2014] (a) 7 (b) 21 (c) 28 (d) 42
Short Answer
Expert verified
The first term of the geometric progression is 21.
Step by step solution
01
Define Terms of the Geometric Progression
In a geometric progression, the terms can be represented as \( a, ar, ar^2, ar^3, ar^4 \) where \( a \) is the first term and \( r \) is the common ratio. Let's establish these representations clearly.
02
Write the Sum of Terms and Their Reciprocals
The sum of the first 5 terms in the geometric progression is \( S = a + ar + ar^2 + ar^3 + ar^4 \). The sum of their reciprocals is \( T = \frac{1}{a} + \frac{1}{ar} + \frac{1}{ar^2} + \frac{1}{ar^3} + \frac{1}{ar^4} \).
03
Use Ratio Condition
According to the exercise, the ratio \( \frac{S}{T} = 49 \). Substitute the expressions of \( S \) and \( T \) in this equation:\[\frac{a + ar + ar^2 + ar^3 + ar^4}{\frac{1}{a} + \frac{1}{ar} + \frac{1}{ar^2} + \frac{1}{ar^3} + \frac{1}{ar^4}} = 49\]
04
Simplify the Equation
Multiply both the numerator and the denominator by \( a^5 \) to simplify:\[(a + ar + ar^2 + ar^3 + ar^4) (ar^4 + ar^3 + ar^2 + ar + 1) = 49\]That simplifies to:\[(1 + r + r^2 + r^3 + r^4)(r^4 + r^3 + r^2 + r + 1) = 49\]Using the formula for the sum of a geometric progression \( \sum_{k=0}^{n} r^k = \frac{r^{n+1}-1}{r-1} \), substitute and solve for \( r \).
05
Calculate First and Third Term Sum
The sum of the first and third terms is given as 35, i.e., \( a + ar^2 = 35 \). Solve for \( a \) by expressing it in terms of the common ratio obtained from the previous step.
06
Solve for the First Term
Using both conditions, solve for \( a \). Substitute the value of \( r \) from Step 4 into the equation from Step 5 and solve for \( a \).
07
Verify and Conclude
Check the obtained value of \( a \) by substituting back into the conditions to ensure consistency. Verify it fits the given choices.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Common Ratio
In a geometric progression, the common ratio is a critical concept. It is the factor by which each term in the sequence is multiplied to get the next term. If you start with the first term as \( a \), the second term will be \( ar \), where \( r \) is the common ratio. Each term in the sequence is thus derived by multiplying the previous term by \( r \).
- Example: If \( a = 2 \) and \( r = 3 \), the sequence becomes: 2, 6, 18, 54, 162...
- Importance: The value of \( r \) determines the nature of the progression. If \( r > 1 \), the terms increase. If \( 0 < r < 1 \), they decrease. A negative \( r \) would result in an alternating sign sequence.
Sum of Terms
Calculating the sum of terms in a geometric series is often required. The formula for the sum of the first \( n \) terms is:
\[ S_n = a \frac{r^n - 1}{r-1} \]
This formula assumes \( r eq 1 \). For our specific case of five terms in a geometric progression, the sum \( S \) becomes:
\[ S = a + ar + ar^2 + ar^3 + ar^4 \]
The rapid growth or decay of the sum slightly depends on \( r \):
\[ S_n = a \frac{r^n - 1}{r-1} \]
This formula assumes \( r eq 1 \). For our specific case of five terms in a geometric progression, the sum \( S \) becomes:
\[ S = a + ar + ar^2 + ar^3 + ar^4 \]
The rapid growth or decay of the sum slightly depends on \( r \):
- If \( r > 1 \), the sequence grows fast, and so does the sum.
- If \( 0 < r < 1 \), the terms become smaller, and the sum approaches a finite limit.
Reciprocal of Series
The reciprocal of a series turns each term into its reciprocal. In a geometric progression, we form another series by taking the reciprocal of each term. If we start with \( a, ar, ar^2, ... \), their reciprocals are \( \frac{1}{a}, \frac{1}{ar}, \frac{1}{ar^2}, ... \)
Next, compute the sum \( T \) of these reciprocal terms for our specific case:
\[ T = \frac{1}{a} + \frac{1}{ar} + \frac{1}{ar^2} + \frac{1}{ar^3} + \frac{1}{ar^4} \]
Next, compute the sum \( T \) of these reciprocal terms for our specific case:
\[ T = \frac{1}{a} + \frac{1}{ar} + \frac{1}{ar^2} + \frac{1}{ar^3} + \frac{1}{ar^4} \]
- Inverse behavior: When the original series grows, the reciprocal series diminishes, and vice versa.
- Benefit: Understanding reciprocal series aids in solving problems involving inverse relationships in natural sciences.
First Term Calculation
Calculating the first term \( a \) in a geometric series often requires using the conditions given in a problem effectively. In our exercise, we had two key conditions:
The techniques used in finding \( a \) can apply to many real-world problems, whether finding initial quantities or predicting unknown first elements in a series.
- The ratio of the sum of the first 5 terms to the sum of their reciprocals is 49.
- The sum of the first and third term is 35.
The techniques used in finding \( a \) can apply to many real-world problems, whether finding initial quantities or predicting unknown first elements in a series.
