Finding the 19th Term of an Arithmetic Progression (A.P.)

The question asks us to find the 19th term of the given Arithmetic Progression (A.P.): 10, 7, 4, …

Understanding the Given A.P.

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.

In the given A.P.: 10, 7, 4, …

• The first term, denoted as $a_1$, is 10.
• To find the common difference, denoted as $d$, we subtract any term from its subsequent term.

Let's calculate the common difference:

• $d = \text{Second term} - \text{First term} = 7 - 10 = -3$
• $d = \text{Third term} - \text{Second term} = 4 - 7 = -3$

The common difference $d$ is indeed -3.

Formula for the nth Term of an A.P.

The formula to find the $n$th term of an Arithmetic Progression is:

\( a_n = a_1 + (n-1)d \)

Where:

• \( a_n \) is the \(n\)th term
• \( a_1 \) is the first term
• \( n \) is the term number (in this case, we want the 19th term, so \(n=19\))
• \( d \) is the common difference

Calculating the 19th Term

We need to find the 19th term, so \(n=19\). We have \(a_1 = 10\) and \(d = -3\). Substituting these values into the formula:

\( a_{19} = a_1 + (19-1)d \)

\( a_{19} = 10 + (18)(-3) \)

Now, perform the multiplication:

\( 18 \times -3 = -54 \)

Substitute this back into the equation for \(a_{19}\):

\( a_{19} = 10 + (-54) \)

\( a_{19} = 10 - 54 \)

Finally, perform the subtraction:

\( a_{19} = -44 \)

So, the 19th term of the A.P. 10, 7, 4, ... is -44.

Revision Table: Key A.P. Concepts

Additional Information: Arithmetic Series

Related to an Arithmetic Progression is an Arithmetic Series, which is the sum of the terms of an A.P.

The sum of the first \(n\) terms of an A.P., denoted as \(S_n\), can be calculated using the following formulas:

1. If the first term (\(a_1\)) and the \(n\)th term (\(a_n\)) are known:

   \( S_n = \frac{n}{2} (a_1 + a_n) \)

2. If the first term (\(a_1\)) and the common difference (\(d\)) are known:

   \( S_n = \frac{n}{2} [2a_1 + (n-1)d] \)

These formulas are useful for finding the sum of a certain number of terms in an A.P.