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Chapter 18: Problem 725

Mean of sequence \(1,2,4,8,16 \ldots \ldots .2^{n-1}\) is (a) \(\left[\left(2^{n}-1\right) / n\right]\) (b) \(\left[\left(2^{n-1}-1\right) /(n-1)\right]\) (c) \(\left[\left(2^{n}+1\right) / n\right]\) (d) \(\left[\left(2^{n}-1\right) /(n-1)\right]\)

Short Answer

Expert verified
The short answer is: \(\frac{2^n - 1}{n}\)

Step by step solution

01

The given sequence is \(1, 2, 4, 8, 16, ... , 2^{n-1}\). This is a geometric sequence with the first term \(a_1 = 1\) and a common ratio \(r = 2\). Let's summarize what we have: First term: \(a_1 = 1\) Common ratio: \(r = 2\) Number of terms: \(n\) #Step 2: Finding the sum of the geometric sequence#

In order to find the mean of the sequence, we need to find the sum of the sequence first. We use the formula for the sum \(S_n\) of a geometric sequence: \[S_n = \frac{a_1(r^n - 1)}{r - 1}\] Now, plug in the values for \(a_1\) and \(r\) that we found in Step 1: \[S_n = \frac{1(2^n - 1)}{2 - 1} = 2^n - 1\] #Step 3: Calculating the mean of the sequence#
02

Now that we have found the sum of the sequence, we can calculate the mean. The mean of a sequence is the sum divided by the number of terms. In this case: Mean = \(\frac{S_n}{n} = \frac{2^n - 1}{n}\) #Step 4: Comparing the calculated mean to the provided options#

We have calculated the mean of the sequence to be \(\frac{2^n - 1}{n}\). Now, let's compare this result to the provided options: (a) \(\left[\left(2^{n}-1\right) / n\right]\) (b) \(\left[\left(2^{n-1}-1\right) /(n-1)\right]\) (c) \(\left[\left(2^{n}+1\right) / n\right]\) (d) \(\left[\left(2^{n}-1\right) /(n-1)\right]\) Our calculated mean matches with option (a). Therefore, the correct answer is: (a) \(\left[\left(2^{n}-1\right) / n\right]\)

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Most popular questions from this chapter

A, B, C can hit the target with probability \((1 / 2),(1 / 3),(1 / 4)\) respectively. What is the probability that exactly two of them can hit the target? (a) \((1 / 2)\) (b) \((1 / 3)\) (c) \((1 / 4)\) (d) \((1 / 5)\)

If standard deviation of 3xi \(-2\) is 8 then variance of \((2 / 3) x i\) is (a) \((144 / 81)\) (b) \((81 / 144)\) (c) \((16 / 9)\) (d) \((4 / 3)\)

If the mean of numbers \(20+x, 24+x, 82+x, 100+x\) \(149+x\) is 75 then the mean of \(130+x, 126+x, 68+x\) \(50+x\) and \(1+x\) is (a) 75 (b) 76 (c) 73 (d) 70

There are 4 addressed covers and 4 letters. If 4 letters are put in 4 covers randomly then the probability that not more than one letter is put in proper cover is (a) \((15 / 24)\) (b) \((7 / 24)\) (c) \((17 / 24)\) (d) \((7 / 17)\)

Using \(1,2,3,4,5,6\) four digit numbers without repetition of any digit are formed. If one number is taken from these what is the probability that the selected number is divisible by \(4 ?\) (a) \((96 / 6 !)\) (b) \(\left(96 /{ }^{6} \mathrm{P}_{4}\right)\) (c) \(\left(84 /{ }^{6} \mathrm{P}_{4}\right)\) (d) None
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