if a3= 17 & a8=37 are terms in annarithmetic sequece, fina a1 & d. the answer is a1=9 & d=4 but i don't understand how they got the answer, please help!

term(3) = a + 2d (by definition)

= 17
term(8) = a + 7d = 37

a+7d = 37
a+2d = 17
subtract them
5d = 20
d = 4
sub into a + 2d=17
a + 8 = 17
d = 9

D1 C3

To find the first term, a1, and the common difference, d, in an arithmetic sequence, we can use the formula for the nth term of an arithmetic sequence.

The formula for the nth term of an arithmetic sequence is given by:
an = a1 + (n - 1) * d

We are given two terms: a3 = 17 and a8 = 37. Using the formula, we can write the equations:
a3 = a1 + (3 - 1) * d
a8 = a1 + (8 - 1) * d

Substitute the given values into the equations:
17 = a1 + 2d
37 = a1 + 7d

Now, we have a system of two equations with two variables (a1 and d). We can solve this system to find the values of a1 and d.

To eliminate a1, let's subtract the first equation from the second equation:
37 - 17 = (a1 + 7d) - (a1 + 2d)
20 = 5d

Solve for d:
d = 20 / 5
d = 4

Substitute the value of d back into the first equation to solve for a1:
17 = a1 + 2(4)
17 = a1 + 8
a1 = 17 - 8
a1 = 9

Therefore, the first term, a1, is 9 and the common difference, d, is 4 in the arithmetic sequence.

To find the values of a1 (the first term) and d (the common difference) in an arithmetic sequence, we need to use the formula for the nth term of an arithmetic sequence.

The formula for the nth term of an arithmetic sequence is given by:

an = a1 + (n-1)d

Where:
an = the value of the nth term
a1 = the value of the first term
d = the common difference between consecutive terms
n = the position of the term in the sequence

In this case, we are given two terms, a3 and a8. Let's use these values to find a1 and d.

First, let's determine the value of a3:
a3 = a1 + (3-1)d -- Equation (1)

Similarly, let's determine the value of a8:
a8 = a1 + (8-1)d -- Equation (2)

Substituting the given values from your question, we have:
a3 = 17
a8 = 37

Using Equation (1):
17 = a1 + (3-1)d
17 = a1 + 2d -- Equation (3)

Using Equation (2):
37 = a1 + (8-1)d
37 = a1 + 7d -- Equation (4)

Now, we have a system of equations (Equations 3 and 4) with two unknowns (a1 and d).

To solve this system, we can either eliminate one variable by subtracting one equation from the other or use a method like substitution or matrices. Let's choose the elimination method.

Let's multiply Equation (3) by 7:
7(17) = 7(a1) + 7(2d)
119 = 7a1 + 14d -- Equation (5)

Now, let's subtract Equation (4) from Equation (5):
119 - 37 = 7a1 + 14d - a1 - 7d
82 = 6a1 + 7d -- Equation (6)

We now have one equation (Equation 6) with just two variables: a1 and d.

Given that a1 and d are both integers, we need to find values that satisfy this equation.

We can select different possible values for a1 and d and check if they satisfy the equation.

One way to do this is by testing values that satisfy a1 = 9 and d = 4.

Using Equation (6) with these values:
82 = 6*(9) + 7*(4)
82 = 54 + 28
82 = 82

Since the equation holds true, we have found a valid solution.

Therefore, the values of a1 and d that satisfy the equations are a1 = 9 and d = 4.