second term= mixed fraction 6 1/2 and Tenth term= mixed fraction 18 1/2, determine the common difference and the first term.

Answer 1

To determine the common difference and the first term, we need the formula for the nth term of an arithmetic sequence. The formula is:

An = A1 + (n-1)d,

where An represents the nth term, A1 represents the first term, and d represents the common difference.

Given that the second term is a mixed fraction 6 1/2, we can write it as a proper fraction:

Second term = 6 1/2 = 13/2.

Similarly, the tenth term is a mixed fraction 18 1/2:

Tenth term = 18 1/2 = 37/2.

Using the formula, we have:

An = A1 + (n-1)d

37/2 = A1 + (10-1)d

37/2 = A1 + 9d ---(1)

Also, we know that:

An = A1 + (n-1)d

13/2 = A1 + (2-1)d

13/2 = A1 + d ---(2)

Now we have a system of equations (1) and (2) to solve for the common difference and the first term.

To solve the system, we can subtract equation (2) from equation (1):

37/2 - 13/2 = A1 + 9d - (A1 + d)

24/2 = A1 + 9d - A1 - d

12 = 8d

Dividing both sides of the equation by 8:

d = 12/8

Simplifying,

d = 3/2

Now that we have found the common difference, we can substitute it back into equation (2) to solve for the first term:

13/2 = A1 + (3/2)

Subtracting (3/2) from both sides:

13/2 - 3/2 = A1

10/2 = A1

5 = A1

Therefore, the common difference is 3/2 and the first term is 5.