Answer :
To solve these problems, let's address them one by one:
Find the common difference of the AP:
The nth term of the arithmetic progression (AP) is given by the formula [tex]a_n = 7 - 4n[/tex]. In an AP, the nth term [tex]a_n[/tex] is expressed in terms of a starting point [tex]a[/tex] and a common difference [tex]d[/tex]:
[tex]a_n = a + (n-1)d[/tex]
For this series, [tex]a_1 = 7 - 4(1) = 3[/tex]. Let's express [tex]a_n[/tex] in terms of the first term and the common difference:
[tex]a_n = 3 + (n-1)(-4)[/tex]
Expanding this, we get:
[tex]a_n = 3 - 4n + 4[/tex]
[tex]a_n = 7 - 4n[/tex]The common difference [tex]d[/tex] is the coefficient of [tex]n[/tex] in the expression, which is [tex]-4[/tex].
Therefore, the common difference of the AP is [tex]-4[/tex].
Which term of the AP is zero?
The given AP is 21, 18, 15, ...
To find which term is zero, use the nth term formula for an AP:
[tex]a_n = a + (n-1)d[/tex]
where [tex]a = 21[/tex] and [tex]d = -3[/tex] (since 18 - 21 = -3).
Set [tex]a_n = 0[/tex] and solve for [tex]n[/tex]:
[tex]0 = 21 + (n-1)(-3)[/tex]
[tex]0 = 21 - 3n + 3[/tex]
[tex]0 = 24 - 3n[/tex]
[tex]3n = 24[/tex]
[tex]n = \frac{24}{3}[/tex]
[tex]n = 8[/tex]Thus, the 8th term of the AP is zero.
Therefore, the common difference of the first AP is [tex]-4[/tex], and the 8th term of the second AP is zero.
