19th term of the A.P.: 10, 7, 4, ….. is

Question

Prepp Team · Accepted Answer

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

Concept	Description	Formula
Arithmetic Progression (A.P.)	A sequence where the difference between consecutive terms is constant.
First Term	The initial term of the sequence.	$a_1$
Common Difference	The constant difference between consecutive terms.	$d = a_n - a_{n-1}$
nth Term	The term at a specific position $n$ in the sequence.	$a_n = a_1 + (n-1)d$

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:

If the first term ($a_1$) and the $n$th term ($a_n$) are known:
$ S_n = \frac{n}{2} (a_1 + a_n) $
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.

Prepp Team · Answer

-47

Prepp Team · Answer

-10

Prepp Team · Answer

-35