The sum of the 2nd and 3rd terms of an arithmetic sequence is 27 and the sum of the 4th and 5th terms is 77. Find the common difference of the sequence.

Give your answer as a decimal.

The common difference of the sequence is 12.5.

Given that,

The sum of the 2nd and 3rd terms of an arithmetic sequence is 27,

And the sum of the 4th and 5th terms is 77.

We have to determine,

The common difference of the sequence?

According to the question,

Let d is the common difference of the sequence,

The second term to the arithmetic series is,

And the third term to the arithmetic series is,

The sum of the second and third terms of the sequence is 27.

The fourth term to the arithmetic series is,

And the fifth term to the arithmetic series is,

The sum of the fourth and fifth terms of the sequence is 77.

On solving both the equations,

Subtract equation 1 from equation 2,

Hence, The required common difference of the sequence is 12.5.

For more details refer to the link given below.

link

d = 12.5

Step-by-step explanation:

Second term = a + d

Third term = a + 2d

a + d + a + 2d = 27

2a + 3d = 27

Fourth term = a + 3d

Fifth term = a + 4d

a + 3d + a + 4d = 77

2a + 7d = 77

2a + 3d = 27 (1)

2a + 7d = 77 (2)

Subtract (1) from (2) to eliminate a

7d - 3d = 77 - 27

4d = 50

Divide both sides by 4

d = 50/4

= 12. 5

d = 12.5

Substitute d = 12.5 into (1)

2a + 3d = 27

2a + 3(12.5) = 27

2a + 37.5 = 27

2a = 27 - 37.5

2a = -10.5

a = -10.5 / 2

= -5.25

a = -5.25

the mistake in the pattern is that :

b. the decimal point was moved one too many place values each time

the correct pattern could be made by :

b. moving the decimal point to the right the same numbers of place values as the exponents