Answer :
To solve the problems involving Arithmetic Progressions (AP), we need to use the formula for the n-th term of an AP, which is given by:
[tex]a_n = a_1 + (n-1) \cdot d[/tex]
where:
- [tex]a_n[/tex] is the n-th term,
- [tex]a_1[/tex] is the first term,
- [tex]d[/tex] is the common difference,
- [tex]n[/tex] is the term number.
1. Which term of the AP 3, 8, 13, 18... is 78?
In this sequence:
- [tex]a_1 = 3[/tex]
- [tex]d = 8 - 3 = 5[/tex]
We need to find [tex]n[/tex] for which [tex]a_n = 78[/tex].
Substituting these values into the formula:
[tex]78 = 3 + (n-1) \cdot 5[/tex]
[tex]78 = 3 + 5n - 5[/tex]
[tex]78 = 5n - 2[/tex]
[tex]80 = 5n[/tex]
[tex]n = \frac{80}{5}[/tex]
[tex]n = 16[/tex]
So, the 16th term of this AP is 78.
2. Find the number of terms in the AP: 7, 13, 19... 205
In this sequence:
- [tex]a_1 = 7[/tex]
- [tex]d = 13 - 7 = 6[/tex]
- The last term [tex]l = 205[/tex]
Using the n-th term formula again:
[tex]205 = 7 + (n-1) \cdot 6[/tex]
[tex]205 = 7 + 6n - 6[/tex]
[tex]205 = 6n + 1[/tex]
[tex]204 = 6n[/tex]
[tex]n = \frac{204}{6}[/tex]
[tex]n = 34[/tex]
Therefore, there are 34 terms in this AP.
