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Sequences and Series

Means

Suppose a person receives a 5% increase in the salary this year. In the next year his salary increases by 15%. The average annual percentage increase is 9.88 and not 10.0. Why is it so? You can understand this clearly by studying the geometric mean. Let us study the topic means in detail.

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Arithmetic Mean

The sum of all of the numbers in a list divided by the number of terms in that list gives the arithmetic mean of that list. Let us first understand the concept of arithmetic progression.

In the arithmetic progression, we know that if the three numbers are in AP, that means if a, b and are in AP, then basically the first two terms a and b will have the difference which will be equal to the next two terms b and c.

So we can say, b – a = c – b. Rearranging the terms,
2b = a + c
b =   \( \frac{a + c }{2} \)

So we can say that this term b is the average of the other two terms a and c. This average in arithmetic progression is called the arithmetic mean.

Arithmetic mean = A = \( \frac{S}{N} \)

where A =  arithmetic mean, N = the number of terms and S = the sum of the numbers in the list.

Means

Browse more Topics under Sequences And Series

Multiple Arithmetics Means between Two Given Numbers

Let a and b be the two given numbers and A1, A2……Abe the arithmetic means between them. Then a, A1, A2….An, will be in AP.  If a is the first term, then b will be the (n+2)th term.

Hence, b = a + (n + 2 – 1 )d
d = \( \frac{b-a}{n+1} \)
So, A= a + \( \frac{b-a}{n+1} \),
A=  a + 2 \( \frac{b-a}{n+1} \),
…..
A= a + n \( \frac{b-a}{n+1} \).

Geometric Mean

Geometric mean b of two terms a and c is given by √(ac). If a, b and c are in geometric progression, then the ratio of the two consecutive terms should be equal.

\( \frac{b}{a} \) = \( \frac{c}{b} \)
Or,  b² = ac
b = √(ac)
This means that b is the geometric mean of a and c

Multiple Geometric Means between Two Given Numbers

Let a and b be the two given numbers. Let, G1, G2, G3….Gbe n geometric mean between them.

G= a\( (\frac{b}{a}) \)\( \frac{m}{n+1} \)

Example

Question: Insert 3 numbers between 1 and 256 so that the resulting sequence is GP

Solution: So we are supposed to insert three numbers, say

1   G1   G2   G3   256

Let’s insert these numbers in such a way that all these numbers are in G.P. We know that the general form of a GP is :a1,  a1r,  a1r², a1r³,  a1rn-1,  arn… So, let us first write a = 1,  Gas ar, G2 as ar², and G= ar³. So,

a = 1 and ar= 256
or, r= (4)4
or, r = 4, -4
GP: 1  ar  ar²  ar³  256

Case 1: r = 4
ar = 1 × 4
ar² =  1 × 4²
ar³ = 1 × 4³
GP: 1, 4,  16,  64, 256
Hence, 4, 16 and 64 are the three terms which we can add between 1 and 256.

Case 2: r = -4
ar = 1 × -4
ar² =  1 × -4²
ar³ = 1 ×-4³
GP: 1, -4,  16,  -64, 256
Hence, -4, 16 and -64 are the three terms which we can add between 1 and 256.

Relationship between A.M and G.M

For terms a and b, let A be A.M and B be G.M, then A.M >/= G.M

A = \( \frac{a + b}{2} \),  G = ab

Let us take a = 5 and b = 5. In this case,
A.M = \( \frac{a + b}{2} \) = \( \frac{5 + 5}{2} \) = 5
G.M = √5×5 = 5

Here we see that A.M = G.M. Let us now take  a = 5 and b = 20. In this case,
A.M = \( \frac{a + b}{2} \) = \( \frac{5 + 20}{2} \) = 12.5
G.M = √5×20 = 10
Here we see that A.M > G.M

So either A.M = G.M or A.M > G.M

Solved Examples for You

Question 1: Find the arithmetic mean of the series 1, 3, 5,……(2n-1)

  1. n
  2. 2n
  3. n/2
  4. n-1

Answer: A is the correct option. First term, a =1 and the common difference, d = 3 – 1 = 2. Let 2n – 1 be the kth term. Then, from the general term formula,

ak = a + (k-1)d
2n-1 = 1 + (k-1) 2
k-1 = n-1
or k =n

Now, S= [(k/2)*(2a+ (k-1)d]

And A.M = Sk/k = [2a + (k-1)d]/2

A.M = (2+(n-1)2)/2 = n

Question 2: What is the mean?

Answer: Mean or arithmetic mean is the average of the number. Moreover, it is the ‘central’ value of a set of numbers that add up all the numbers then divided by the total numbers there are.

Question 3: What does “!” mean in math?

Answer: In mathematics, the symbol of exclamation mark that is “!” is used as the factorial of an integer which is the product of all the integers below it down to one. For instance 6! = 6 × 5 × 4 × 3 × 2 × 1 = 720.

Question 4:  Explain what is mean mathematically?

Answer: Mathematically, mean is the average of a set of numbers, which we also know by the name arithmetic mean. Furthermore, to find the mean of a data set, add up all the numbers in the set, and then divide the total by the number of a number set.

Question 5: Which is better mean or median?

Answer: Mean is better because it tells you the different things about the central tendency of a distribution. Moreover, in an ordered distribution (e.g., smallest to largest) the median is just the exact mid-value that is 50% of the distribution above it and 50% of the distribution below it.

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