Answer:
[tex]a_{63}[/tex] = 1
Step-by-step explanation:
The n th term of an arithmetic progression is
[tex]a_{n}[/tex] = a + (n - 1)d
where a is the first term and d the common difference
use the 9 th term and 7 th term to find a and d
[tex]a_{9}[/tex] = a + 8d = [tex]\frac{1}{7}[/tex] → (1)
[tex]a_{7}[/tex] = a + 6d = [tex]\frac{1}{9}[/tex] → (2)
Subtract (2) from (1) term by term
2d = [tex]\frac{2}{63}[/tex] ⇒ d = [tex]\frac{1}{63}[/tex]
Substitute this value into (2) and solve for a
a + [tex]\frac{6}{63}[/tex] = [tex]\frac{1}{9}[/tex]
a = [tex]\frac{1}{9}[/tex] - [tex]\frac{6}{63}[/tex] = [tex]\frac{1}{63}[/tex]
Hence
[tex]a_{63}[/tex] = [tex]\frac{1}{63}[/tex] + [tex]\frac{62}{63}[/tex] = 1
