PROGRESSIONS

Arithmetic Progression (AP) refers to a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is called the "common difference."

The general form of an arithmetic progression is:

$a,a+d,a+2d,a+3d,\dots$

Where:

• $a$ is the first term.
• $d$ is the common difference between terms.

Key Formulas and Concepts:

1. $n$th Term of an AP: The $n$th term of an AP is given by: n​=a+(n−1)⋅d Where Tn​ is the $n$th term, $a$ is the first term, $d$ is the common difference, and $n$ is the term number.

2. Sum of $n$ Terms of an AP (Arithmetic Series): The sum of the first $n$ terms of an AP is given by the formula: S_n​=n/2​(2a+(n−1)⋅d) Where S_n​ is the sum of the first $n$ terms, $a$ is the first term, $d$ is the common difference, and $n$ is the number of terms.

3. $n$th Term from the End: The $n$th term from the end of an AP can be found using: $\text{Term from end}=\text{Last term}-(n-1)\times \text{common difference}$

4. Finding Common Difference: If a_n​ and a_m​ are two terms in an AP, the common difference $d$ can be calculated as:

Properties:

• An AP can have an infinite number of terms.
• The sum of an arithmetic series can be calculated using the formula for the sum of $n$ terms.
• If a_n​ is the last term of an AP, the sum of the series can also be calculated as S_n​=n/2(a+a_n​)

S= n(a+L)/2 L = a + (n – 1) d S = n/2 [2a+(n–1) d]

 

 

Arithmetic Mean

 

The arithmetic mean, often referred to as the average, is a measure used to determine the central tendency of a set of numbers. It's calculated by finding the sum of all values in a dataset and dividing that sum by the total number of values.

The formula to find the arithmetic mean is:

 

Key Points:

1. Calculation: Add up all the values in the dataset and divide the sum by the total number of values.

2. Properties:

   • The sum of deviations of each value from the mean is zero.
   • The mean is affected by extreme values (outliers) in a dataset.

3. Use in Statistics:

   • Provides a central value that represents the entire dataset.
   • Widely used in various statistical analyses and interpretations.

4. Example: For instance, consider a dataset: $4,6,8,10,12$ The sum of these values is $4+6+8+10+12=40$. Dividing by the total number of values ($5$ in this case) gives an arithmetic mean of $40÷5=8$.

 

 

Geometric Progression (GP)

 

Geometric Progression (GP) is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the "common ratio."

The general form of a geometric progression is:

a,a^r,ar²,ar³,…

Where:

• $a$ is the first term.
• $r$ is the common ratio between consecutive terms

Key Formulas and Concepts:

$1.n$th Term of a GP: The $n$th term of a GP is given by:

Tn​=a⋅r⁽ⁿ⁻¹⁾ Where Tn​ is the $n$th term, $a$ is the first term, $r$ is the common ratio, and $n$ is the term number

2.Sum of $n$ Terms of a GP (Geometric Series): The sum of the first $n$ terms of a GP is given by the formula:

                                                  Sn​=a⋅^(rn−1)​/(r-1)

Where Sn​ is the sum of the first $n$ terms, $a$ is the first term, $r$ is the common ratio, and $n$ is the number of terms

3.Infinite Geometric Series:

If $|r|<1$, an infinite geometric series converges to a finite value. The sum of an infinite GP is given by: S=a/1−r​ Where $S$ is the sum of an infinite series

Properties:

• A GP can have an infinite number of terms.
• If $|r|>1$, the series diverges and does not have a finite sum.
• $r$ can be positive or negative, leading to different sequences (increasing or decreasing)

Sn=a(r^n-1)/(r-1) If r > 1, then Sn=a({1-r}^n )/(1-r) If r< 1, then

 

Geometric Mean

 

The geometric mean is a measure of central tendency used to find the average of a set of numbers that are multiplied together. Unlike the arithmetic mean that adds values and divides by the count, the geometric mean finds the "nth root" of the product of $n$ values.

The formula to find the geometric mean for $n$ values x₁​,x₂​,x₃​,…,x_n​ is:

Geometric Mean  =

Key Points:

1. Calculation: Multiply all the values together and take the $n$th root, where $n$ is the number of values.

2. Use in Context:

   • Utilized when dealing with rates of growth, ratios, or values that multiply together (e.g., compound interest rates).
   • Often used for calculating average growth rates or returns over multiple periods.

3. Properties:

   • Useful when dealing with percentage changes or ratios.
   • It is sensitive to extreme values (outliers) in the dataset.

4. Example: For example, consider a dataset: $2,4,8,16$ The product of these values is $2\times 4\times 8\times 16=1024$. Taking the 4th root of 1024 gives a geometric mean of $4$

 

 

$Practice\; Questions\; on\; Progressions$   Arithmetic Progression (AP): Question 1: The sum of the first $n$ terms of an AP is given by Sn​=3n2+5n. What is the $r$th term of the AP? A) $3r+5$ B) 3r2+5r C) 3r2+5 D) 3r+5r2 Question 2: The $4$th term of an AP is $12$ and the $10$th term is $28$. Find the sum of the first $15$ terms of this AP. A) $375$ B) $360$ C) $390$ D) $400$ Geometric Progression (GP): Question 3: The sum of the first $n$ terms of a GP is given by Sn​=5(2n−1). What is the $r$th term of the GP? A) 10r B) 2r+1 C) 5⋅2r D) 5⋅2r−1 Question 4: In a GP, the first term is $3$ and the fourth term is 818881​. Find the sum of the first $6$ terms of this GP. A) 938893​ B) 638863​ C) 18388183​ D) 12388123​ Answers: C) 3r2+5 C) 390 C) 5⋅2r D) 123/8

 

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