A geometric progression (GP) consists of 200 terms. If the sum of odd terms of the GP is m, and the sum of even terms of the GP is n, then what is its common ratio?
n/m
Understanding the Geometric Progression Problem
This problem asks us to find the common ratio of a geometric progression (GP) with 200 terms. We are given the sum of the terms that appear at odd positions and the sum of the terms that appear at even positions. Let's break down the problem and use the properties of GPs to find the common ratio.
A geometric progression is a sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. If the first term is \(a\) and the common ratio is \(r\), the terms of the GP are \(a, ar, ar^2, ar^3, \dots\).
Setting up the Geometric Progression
The GP has 200 terms. Let the first term be \(a\) and the common ratio be \(r\). The terms are:
\(T_1 = a\)
\(T_2 = ar\)
\(T_3 = ar^2\)
\(T_4 = ar^3\)
...
\(T_{200} = ar^{199}\)
Sum of Odd Terms in the GP
The odd terms are the terms at positions 1, 3, 5, ..., up to 199. These terms are:
\(T_1, T_3, T_5, \dots, T_{199}\)
Substituting the terms:
\(a, ar^2, ar^4, \dots, ar^{198}\)
This sequence itself forms a geometric progression.
- The first term of this new GP is \(A = a\).
- The common ratio of this new GP is \(R = \frac{ar^2}{a} = r^2\).
- To find the number of terms, we look at the powers of \(r\): \(r^0, r^2, r^4, \dots, r^{198}\). The powers are \(2k\) for \(k=0, 1, 2, \dots, 99\). So there are 99 - 0 + 1 = 100 terms.
The sum of these odd terms is given as \(m\). Using the formula for the sum of a GP, \(S_N = A \frac{R^N - 1}{R - 1}\), where \(N=100\):
\(m = a \frac{(r^2)^{100} - 1}{r^2 - 1} = a \frac{r^{200} - 1}{r^2 - 1}\)
This equation is valid as long as \(r^2 \neq 1\).
Sum of Even Terms in the GP
The even terms are the terms at positions 2, 4, 6, ..., up to 200. These terms are:
\(T_2, T_4, T_6, \dots, T_{200}\)
Substituting the terms:
\(ar, ar^3, ar^5, \dots, ar^{199}\)
This sequence also forms a geometric progression.
- The first term of this new GP is \(A' = ar\).
- The common ratio of this new GP is \(R' = \frac{ar^3}{ar} = r^2\). This is the same common ratio as the GP of odd terms.
- To find the number of terms, we look at the powers of \(r\): \(r^1, r^3, r^5, \dots, r^{199}\). The powers are \(2k+1\) for \(k=0, 1, 2, \dots, 99\). So there are 99 - 0 + 1 = 100 terms.
The sum of these even terms is given as \(n\). Using the formula for the sum of a GP, \(S_N = A' \frac{R'^N - 1}{R' - 1}\), where \(N=100\):
\(n = ar \frac{(r^2)^{100} - 1}{r^2 - 1} = ar \frac{r^{200} - 1}{r^2 - 1}\)
This equation is valid as long as \(r^2 \neq 1\).
Finding the Common Ratio
We have two equations from the sums:
1) \(m = a \frac{r^{200} - 1}{r^2 - 1}\)
2) \(n = ar \frac{r^{200} - 1}{r^2 - 1}\)
We want to find \(r\). Notice that the expression \(\frac{r^{200} - 1}{r^2 - 1}\) appears in both equations.
Let's divide the second equation by the first equation (assuming \(m \neq 0\) and \(\frac{r^{200} - 1}{r^2 - 1} \neq 0\)):
\(\frac{n}{m} = \frac{ar \frac{r^{200} - 1}{r^2 - 1}}{a \frac{r^{200} - 1}{r^2 - 1}}\)
The term \(\frac{r^{200} - 1}{r^2 - 1}\) cancels out from the numerator and the denominator, as does the first term \(a\) (assuming \(a \neq 0\), which is true for a GP).
\(\frac{n}{m} = r\)
Thus, the common ratio \(r\) of the original geometric progression is \(\frac{n}{m}\).
Verification and Conclusion
The common ratio of the GP is indeed the ratio of the sum of even terms to the sum of odd terms. Let's summarize:
| Quantity | Symbol | Description |
|---|---|---|
| Sum of Odd Terms | \(m\) | Sum of \(T_1, T_3, \dots, T_{199}\) |
| Sum of Even Terms | \(n\) | Sum of \(T_2, T_4, \dots, T_{200}\) |
| Common Ratio | \(r\) | The number multiplying each term to get the next |
We found that \(r = \frac{n}{m}\).
Revision Table: Geometric Progression Concepts
| Concept | Formula/Description | Notes |
|---|---|---|
| General Term of GP | \(T_k = ar^{k-1}\) | \(a\): first term, \(r\): common ratio, \(k\): term number |
| Sum of N terms of GP | \(S_N = a \frac{r^N - 1}{r - 1}\) | Valid when \(r \neq 1\). If \(r=1\), \(S_N = Na\). |
| GP of Odd Terms | \(a, ar^2, ar^4, \dots\) | First term: \(a\), Common ratio: \(r^2\) |
| GP of Even Terms | \(ar, ar^3, ar^5, \dots\) | First term: \(ar\), Common ratio: \(r^2\) |
Additional Information: Properties of Geometric Progressions
Geometric progressions are important sequences in mathematics with many interesting properties and applications.
- Product of Terms: The product of the first \(N\) terms of a GP with first term \(a\) and common ratio \(r\) is \(P_N = a^N r^{N(N-1)/2}\).
- Infinite GP Sum: If the absolute value of the common ratio \(|r| < 1\), the sum of an infinite geometric progression converges to \(S_\infty = \frac{a}{1 - r}\).
- Relationship between terms: Any term squared is equal to the product of the terms equidistant from it, e.g., \((ar^k)^2 = (ar^{k-j})(ar^{k+j})\).
- Applications: GPs are used in calculating compound interest, modeling population growth, radioactive decay, and in various areas of physics and engineering.
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