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

Final answer:

To find the values of a and b in an arithmetic sequence, we can set up a system of equations using the given terms. Solving the system will give us the values of a and b.

Explanation:

Let's use the information given to find the values of a and b. We can set up a system of equations using the first four terms of the arithmetic sequence.

1. The first term is 2, so we know that: a-b = 2.

2. The second term is a-b, so we can write: a-b + d = 2a+b+7, where d represents the common difference.

3. The third term is 2a+b+7, so we have: a-b + 2d = 2a+b+7 + d.

4. The fourth term is a-3b, so we get: a-b + 3d = a-3b + 2d.

We can solve this system of equations to find the values of a and b.

After simplifying and solving the system, we find that a = 10 and b = 8.

Learn more about Arithmetic sequence here:

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