What is the 22nd term in the arithmetic sequence in which a4 is 73 and a10 is -11?

Answer 1

In an arithmetic sequence, the ( n )-th term can be expressed as ( a_n = a_1 + (n-1)d ), where ( a_1 ) is the first term and ( d ) is the common difference. Given ( a_4 = 73 ) and ( a_{10} = -11 ), we can set up the equations:

1. ( a_1 + 3d = 73 )
2. ( a_1 + 9d = -11 )

Subtracting the first equation from the second gives ( 6d = -84 ), so ( d = -14 ). Substituting ( d ) back into the first equation yields ( a_1 = 73 - 3(-14) = 73 + 42 = 115 ).

To find ( a_{22} ), we use ( a_{22} = a_1 + 21d = 115 + 21(-14) = 115 - 294 = -179 ). Thus, the 22nd term is (-179).