The 6th term of AP is 17 and 13th term is 38. Determine the 19th term?

the two terms are 7 apart, so

7d = 38-17 = 21
d = 3
The 19th term a_19 = a_13 + 6d = 38+6*3 = 56

Well, to determine the 19th term of this arithmetic progression (AP), let me first entertain you with a little joke:

Why don't scientists trust atoms?
Because they make up everything!

Now, let's get back to the question. We can use the formula to find an arithmetic progression term:

An = A1 + (n - 1)d

Given that the 6th term (A6) is 17 and the 13th term (A13) is 38, we can plug in those values and solve for the common difference (d).

17 = A1 + (6 - 1)d
38 = A1 + (13 - 1)d

After some calculations (assuming you're not afraid of numbers), we can find that A1 is 2 and d is 3.

Now, we can use the formula to find the 19th term (A19):

A19 = A1 + (19 - 1)d

A19 = 2 + (19 - 1)3

A19 = 2 + 18 * 3

A19 = 2 + 54

A19 = 56

So, the 19th term of this arithmetic progression is 56.

To determine the 19th term of the arithmetic progression (AP), we need to find the common difference, d, first.

Given that the 6th term is 17 and the 13th term is 38, we can use these two terms to find the common difference.

The formula to find the nth term of an arithmetic progression is:

an = a1 + (n - 1) * d

Where:
an = nth term
a1 = 1st term
n = position of the term
d = common difference

Using the 6th term, we have:

17 = a1 + (6 - 1) * d

Simplifying this equation, we get:

17 = a1 + 5d

Next, using the 13th term, we have:

38 = a1 + (13 - 1) * d

Simplifying this equation, we get:

38 = a1 + 12d

We now have a system of two equations:

17 = a1 + 5d
38 = a1 + 12d

We can solve this system of equations to find the values of a1 and d.

To determine the 19th term of the arithmetic progression (AP), we need to find the common difference (d) first.

We are given the 6th term (a6 = 17) and 13th term (a13 = 38).

Using the formula for the nth term of an AP:
an = a1 + (n-1)d

We can set up two equations to solve for the common difference.

For the 6th term:
17 = a1 + (6 - 1)d
17 = a1 + 5d

For the 13th term:
38 = a1 + (13 - 1)d
38 = a1 + 12d

Next, we can solve these two equations simultaneously to find the values of a1 and d.

We can start by subtracting the first equation from the second equation:
(38 - 17) = (a1 + 12d) - (a1 + 5d)
21 = 12d - 5d
21 = 7d

Dividing both sides by 7, we get:
d = 3

Now, we can substitute the value of d back into either of the original equations to solve for a1. Let's use the first equation:

17 = a1 + 5(3)
17 = a1 + 15
a1 = 17 - 15
a1 = 2

Therefore, the first term (a1) is 2 and the common difference (d) is 3.

Now we can use the formula for the nth term of an AP to find the 19th term (a19):

a19 = a1 + (19 - 1)d
a19 = 2 + (19 - 1)(3)
a19 = 2 + 18(3)
a19 = 2 + 54
a19 = 56

Therefore, the 19th term of the AP is 56.