The mean of the series 1, 2, 4, 8...., 2n is given as:
Calculating the Mean of a Geometric Series
The problem asks us to find the mean of the series 1, 2, 4, 8, ..., 2ⁿ.
Understanding the Given Series
Let's examine the terms of the series:
- The first term is 1.
- The second term is 2.
- The third term is 4.
- The fourth term is 8.
- ...
- The last term is 2ⁿ.
We can observe a pattern: each term is obtained by multiplying the previous term by 2. This type of series is known as a geometric series.
Identifying the Key Components of the Geometric Series
To find the mean, we need the sum of the terms and the number of terms.
- The first term, often denoted as 'a', is 1.
- The common ratio, often denoted as 'r', is the ratio of any term to its preceding term (e.g., 2/1 = 2, 4/2 = 2). So, r = 2.
- The terms can be written as 2⁰, 2¹, 2², 2³, ..., 2ⁿ.
- The exponents range from 0 to n. This means there are \(n - 0 + 1 = n + 1\) terms in the series.
So, we have a geometric series with:
- First term (a) = 1
- Common ratio (r) = 2
- Number of terms (N) = n + 1
Calculating the Sum of the Series
The sum (S) of a geometric series with first term 'a', common ratio 'r', and N terms is given by the formula:
\(S = \frac{a(r^N - 1)}{r - 1}\)
In our case, a = 1, r = 2, and N = n + 1. Substituting these values into the formula:
\(S = \frac{1(2^{n+1} - 1)}{2 - 1}\)
\(S = \frac{2^{n+1} - 1}{1}\)
\(S = 2^{n+1} - 1\)
The sum of the series 1, 2, 4, ..., 2ⁿ is \(2^{n+1} - 1\).
Calculating the Mean of the Series
The mean of a series is the sum of the terms divided by the number of terms.
\(\text{Mean} = \frac{\text{Sum of terms}}{\text{Number of terms}}\)
We found the sum of terms to be \(2^{n+1} - 1\), and the number of terms is \(n + 1\).
\(\text{Mean} = \frac{2^{n+1} - 1}{n + 1}\)
Comparing with the Options
Let's look at the given options:
- \(\rm \frac{2^{n-1}-1}{n+1}\)
- \(\rm \frac{2^{n+1}-1}{n}\)
- \(\rm \frac{2^{n+1}-1}{n+1}\)
- \(\rm \frac{2^{n-1}}{n}\)
Our calculated mean, \(\frac{2^{n+1} - 1}{n + 1}\), matches option 3.
Conclusion
The mean of the series 1, 2, 4, 8, ..., 2ⁿ is \(\frac{2^{n+1} - 1}{n + 1}\).
Revision Table: Key Concepts
| Concept | Description | Formula/Example |
|---|---|---|
| Geometric Series | A sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. | a, ar, ar², ar³, ... |
| First Term (a) | The initial term of the series. | In 1, 2, 4, ..., 2ⁿ, a = 1 |
| Common Ratio (r) | The constant factor between consecutive terms. | In 1, 2, 4, ..., 2ⁿ, r = 2/1 = 2 |
| Number of Terms (N) | The total count of terms in the series. | For 2⁰, 2¹, ..., 2ⁿ, N = n+1 |
| Sum of Geometric Series | The total value obtained by adding all terms. | \(S_N = \frac{a(r^N - 1)}{r - 1}\) (if r ≠ 1) |
| Mean (Average) | The sum of a set of values divided by the number of values. | Mean = \(\frac{\text{Sum}}{\text{Count}}\) |
Additional Information: Exploring Series and Mean
Understanding different types of series and how to calculate their sum and mean is fundamental in mathematics and statistics.
- Arithmetic Series: In contrast to a geometric series, an arithmetic series has a constant difference between consecutive terms (e.g., 3, 5, 7, 9...). The sum of an arithmetic series is given by \(S_N = \frac{N}{2}(2a + (N-1)d)\), where 'a' is the first term and 'd' is the common difference.
- Relationship between Sum and Mean: The formula for the mean directly depends on the sum and the number of terms. If you can calculate the sum and count the terms, you can find the mean.
- Applications: Geometric series appear in various fields, including finance (compound interest), population growth, and physics (radioactive decay). Calculating their mean helps understand the average value over a period or across terms.
Always identify the type of series (arithmetic, geometric, etc.) before applying summation formulas. Correctly determining the number of terms is also crucial for both sum and mean calculations.
Similar Questions
Which of the following is a merit of data tabulation?
In seasonal variations, the duration of time is not more than:
If a constant is added to each observation of a data set, then which of the following measures of dispersion will change?
Which of the following is NOT true for seasonal variation?
The analysis of variance technique was developed by:
For the series 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, the minimum possible value of \(\rm \Sigma_{i=1}^n(x_i-A)^2\) can be attained at:
Suppose that a sample of 100 independent draws from a normal distribution having unknown mean μ and known variance σ2 = 1 is observed. If the sample mean is 5, then the 95% confidence interval for μ is:
Consider the following ANOVA table.
Source of variation Degrees of freedom The sum of Squares (SS) Mean SS F Ratio Treatments a b c 5 Error 12 d 20 Total 15 540
The values of a, b, c and d are, respectively:If the mean, mode and quartile deviation of a distribution is 2, 7 and 3, respectively, then Karl Pearson's coefficient of skewness is given by:
The component containing the overall upward or downward pattern of the data in an annual time series is:
Important Questions from Statistics
Methods of presentation of data are:
Consider 35 i.i.d. observations X1, X2, ...., X15 and Y1, Y2, ..., Y20. Let R be the Wilcoxon's rank sum statistic based on the ranks of the X' s in the combined sample. Then the expected value of R is
Let I, J > 5. Consider two-way ANOVA, where the observations satisfy the linear model
yij = α + βi + γj + εij, 1 ≤ i ≤ I, 1 ≤ j ≤ J
E(εij) = 0, Var(εij) = σ2, \(\sum_{i=1}^{!} β_i=\sum_{j=1}^j γ_j=0 .\)
In this set - up
Let X1,X2,...,Xn be a random sample from exponential distribution with mean θ. Which of the following is NOT a sufficient statistic for θ?
A newly developed algorithm for random number generation needs to be tested. The first step is to check whether the sequence of numbers generated can be considered a random sample from the uniform distribution on the interval (0, 1). Which of the following is an appropriate nonparametric test?
