algebra 2 need help. sequences and series.

Question

What is the rule for the nth term of the arithmetic sequence that has the two given terms.

11.) a12=0, a19=28

12.) d=-5, a10=-60

Answer 1

The nth term of the arthmetic sequance = t n = a + ( n - 1 ) d.

a = first term and  d = common difference.

1 )  a12 = 0  , a19 = 28 .

12 th term = 0

a + ( 12 - 1 ) d = 0

a + 11 d = 0 -------> ( 1 )

19 th term = 28

a + ( 19 - 1 ) d =28

a + 18 d = 28  -------> ( 2 )

Subtracting the eq (1 ) from eq ( 2 ).

( a + 18 d ) -   ( a + 11 d ) =  28 - 0

a + 18 d - a - 11 d  = 28

7 d = 28

d = 28 / 7 ⇒ d = 4 .

Substitute d  value in eq ( 1 )

a + 11 d = 0

a + 11( 4 )  = 0 ⇒ a + 44 = 0 ⇒ a = - 44.

Therefore a = - 44 and d = 4.

2 )  d = - 5 , a10 = - 60

t n = a + ( n - 1 ) d.

10th term = - 60

a + ( 10 - 1 ) d = - 60

a + 9 d = - 60 -------> ( 1 )

 d = - 5

Substitute d value in eq ( 1 )

a + 9 d = - 60

a + 9 ( - 5 )  = - 60 ⇒ a -  45 = - 60 ⇒ a = - 60 + 45 = - 15

a = - 15.

Therefore a = - 15.