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].
