Q1 : If a, b, c are in Arithmetic Progression (AP), then (a-b) / (b-c) 

Answer : 1 

Explanation : if a,b,c are in AP then (a-b) = -d ( where d is common difference) , and (b-c) = -d 

hence -d/-d = 1

Q2: Find the common difference of AP, whose nth term is an = 5n+3

Answer : If an=5n+1, 

  when  n = 1,    a1 = 5(1)+1 = 6

            n = 2,    a2  = 5(2)+1 = 11 

common difference d = a2-a1 = 11-6 = 5

 

Q3 :  Find four consecutive terms in AP whose sum is 20 and sum of whose squares is 120.

Answer :  Consider 4 consecutive terms of AP as a-3d, a-d, a+d and a+3d 

Sum, (a-3d)+(a-d)+(a+d)+(a+3d) = 20 

         4a = 20 

        a = 5

(a-3d)² + (a-d)² + (a+d)² + (a+3d)² = 120

expanding,

a² -6ad+9d² + a² -2ad+d² +a² +2ad+d² +a² +6ad+9d²= 120

4a² + 20d² = 120

4(5)² + 20d² = 120

100 + 20d² = 120

 20d² = 20

 d² =1

d = ±1

with a =5 and d = 1, AP is 

(a-3d) = 5-3(1) = 2

(a-d) = 5-1 = 4

(a+d) = 5+1 = 6

(a+3d) = 5+3(1) = 8

Hence the Answer : 2,4,6,8