1. What is the twenty-fifth term of the arithmetic sequence with a1 = –1 and d = –10? 

Substitute a1=-1, d=-10, and n=25 in

an = a1 + (n-1)d

Answer: -241 


2.What is the twenty-ninth term of arithmetic sequence with a1 = 13 and d = –5/2? 

Substitute a1=13, d=-5/2, and n=29 in

an = a1 + (n-1)d

Answer: -57 

3. What are the two arithmetic means between –13 and 8? 

a1 - -13, a2 = ?, a3 = ?, a4 = 8

Each term differs from the preceding term by d.

a2 is a1 + d, and a3 is a4 - d, so

a1 = -13, a2 = -13+d, a3 = 8-d, a4 = 8
 
 a2 + d = a3
-13+d+d = 8-d
 -13+2d = 8-d
     3d = 21
      d = 7


 a2 = -13+7 = -6
 a3 = 8-7 = 1

-13, -6, 1, 8

They are -6 and 1



4. What is Sn for the arithmetic series with d = –4, an = 27, and n = 9? 

First substitute an = 27, d=-4, and n=9 in

an = a1 + (n-1)d

and solve for a1.

Get a1 = 59

Substitute a1 = 59, d = -4, leave n as just n in

Sn = (n/2)[a1 + (n-1)d]

Simplify

Get

Sn = n(61-2n)

Edwin