Answer :

Certainly! Let's find the 20th term ([tex]\(a_{20}\)[/tex]) of an arithmetic progression (AP) given the first term ([tex]\(a = 10\)[/tex]) and the common difference ([tex]\(d = -3\)[/tex]).

In an arithmetic progression, the [tex]\(n\)[/tex]-th term ([tex]\(a_n\)[/tex]) can be calculated using the formula:

[tex]\[ a_n = a + (n - 1) \cdot d \][/tex]

Here:
- [tex]\(a\)[/tex] is the first term.
- [tex]\(d\)[/tex] is the common difference.
- [tex]\(n\)[/tex] is the term number.
- [tex]\(a_n\)[/tex] is the [tex]\(n\)[/tex]-th term we want to find.

We need to find the 20th term ([tex]\(a_{20}\)[/tex]), so [tex]\(n = 20\)[/tex].

Let's substitute the values into the formula:

[tex]\[ a_{20} = a + (20 - 1) \cdot d \][/tex]

Firstly, calculate [tex]\(20 - 1\)[/tex]:

[tex]\[ 20 - 1 = 19 \][/tex]

Now, substitute [tex]\(a = 10\)[/tex] and [tex]\(d = -3\)[/tex]:

[tex]\[ a_{20} = 10 + 19 \cdot (-3) \][/tex]

Next, calculate the product [tex]\(19 \cdot (-3)\)[/tex]:

[tex]\[ 19 \cdot (-3) = -57 \][/tex]

Finally, add this result to the first term [tex]\(a = 10\)[/tex]:

[tex]\[ a_{20} = 10 + (-57) \][/tex]

[tex]\[ a_{20} = 10 - 57 \][/tex]

[tex]\[ a_{20} = -47 \][/tex]

So, the 20th term [tex]\(a_{20}\)[/tex] of the arithmetic progression is [tex]\(-47\)[/tex].