The first four terms of an arithmetic sequence are 2,a-b, 2a+b+7 and a - 3b, where a and b are constants. Find a and b

Arithmetic Sequence

The  values of a and b is a = 2 and b=3

Step-by-step explanation:

Given the terms of the arithmetic sequence are

2 , a - b ,  2a + b + 7, a - 3b

Let the common difference be D

Therefore,

The difference between the first two consecutive terms is

(a – b) – 2 = D        ( 1 )

The difference between the next two consecutive terms is

D = (2a + b+7) – ( a - b ) (2 )

Equating equation 1 and equation 2

⇒ (a – b) -2 =(2a+b+7)-(a-b)

⇒ a – b – 2 =  a + 2b +7

⇒ 3b = -9

⇒ b = -3

Similarly

The difference between the next two consecutive terms is

D = (a-3b)-(2a+b+7)  (3)

⇒ (a-3b)-(2a+b+7)=(2a+b+7)-(a-b)    

⇒ a-3b)-(2a+b+7) -a  - 4b -7 === a+2b+7

⇒ 2a = - ( 14 + 6b)

⇒  a = -( 7 + 3b)

⇒  a = - ( 7 – 3*3 )

Thus the value of a = 2

Hence , the values of a and b is a = 2 and b=3