Understanding the Arithmetic Progression Problem

The question asks us to find the 20th term of an Arithmetic Progression (AP). We are given the sum of the first 14 terms and the first term of the AP.

Key Concepts for Arithmetic Progression (AP)

An Arithmetic Progression is a sequence of numbers such that the difference between the consecutive terms is constant. This constant difference is called the common difference, denoted by 'd'.

• The first term is usually denoted by 'a'.
• The n-th term of an AP is given by the formula: \(a_n = a + (n-1)d\).
• The sum of the first n terms of an AP is given by the formula: \(S_n = \frac{n}{2}[2a + (n-1)d]\).

Step-by-Step Solution to Find the 20th Term

We are given the following information:

• Sum of the first 14 terms (\(S_{14}\)) = 1050
• First term (a) = 10
• Number of terms (n) for the sum = 14

We need to find the 20th term (\(a_{20}\)). To find \(a_{20}\), we first need to determine the common difference (d) of the AP.

Finding the Common Difference (d)

We can use the formula for the sum of the first n terms of an AP:

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

Substitute the given values into the formula with n=14:

\(S_{14} = \frac{14}{2}[2a + (14-1)d]\)

\(1050 = 7[2(10) + 13d]\)

Now, simplify and solve for d:

\(1050 = 7[20 + 13d]\)

Divide both sides by 7:

\(\frac{1050}{7} = 20 + 13d\)

\(150 = 20 + 13d\)

Subtract 20 from both sides:

\(150 - 20 = 13d\)

\(130 = 13d\)

Divide both sides by 13:

\(d = \frac{130}{13}\)

\(d = 10\)

The common difference of the AP is 10.

Finding the 20th Term (\(a_{20}\))

Now that we have the first term (a=10) and the common difference (d=10), we can find the 20th term using the formula for the n-th term:

\(a_n = a + (n-1)d\)

Substitute a=10, d=10, and n=20 into the formula:

\(a_{20} = 10 + (20-1)10\)

\(a_{20} = 10 + (19)10\)

\(a_{20} = 10 + 190\)

\(a_{20} = 200\)

The 20th term of the Arithmetic Progression is 200.

Summary of Results

Given:

• First term (a) = 10
• Sum of first 14 terms (\(S_{14}\)) = 1050

Calculated:

• Common difference (d) = 10
• 20th term (\(a_{20}\)) = 200

Revision Table: Arithmetic Progression Formulas

Additional Information: Understanding Arithmetic Progressions

Arithmetic Progressions are fundamental sequences in mathematics. They appear in various applications, from simple counting patterns to more complex financial calculations.

• The terms of an AP form a linear sequence when plotted on a graph.
• If the common difference 'd' is positive, the AP is increasing.
• If the common difference 'd' is negative, the AP is decreasing.
• If the common difference 'd' is zero, all terms in the AP are the same.
• Understanding the relationship between the first term, common difference, number of terms, n-th term, and the sum of terms is crucial for solving AP problems.