1. ## Arithmetic progression

The first and third term of arithmetic progression are x and y respectively.Find the sixteenth term in term of x and y.

2. Originally Posted by mastermin346
The first and third term of arithmetic progression are x and y respectively.Find the sixteenth term in term of x and y.
The values of the arithmetic progression are calculated by

$a_n=a_1+(n-1)\cdot d$

with

$d = a_{n+1} - a_n$

In your case $d = \frac{y-x}2$.

Plug in the values you know:

$n = 16$
$a_1 = x$
$a_3 = y$

and calculate $a_{16}$.

3. Originally Posted by mastermin346
The first and third term of arithmetic progression are x and y respectively.Find the sixteenth term in term of x and y.
$x + (x+d) + (x+2d) + (x+3d) + ... (x + 15d)$
where $d$ is the constant difference between each successive term.

In our case, we are told that the third term is y. Thus, $x+2d = y$ .

So, $d = \frac{y-x}{2}$

$a_1 = x$

$a_2 = x + \frac{y-x}{2} = \frac{2x}{2} + \frac{y-x}{2} =
\frac{2x+y-x}{2} = \frac{x+y}{2}
$

$a_3 = \frac{x+y}{2} + \frac{y-x}{2}= \frac{2y}{2} = y$
...
...
...

$a_{16} = x + 15d = x + 15\big{(}\frac{y-x}{2}\big{)}$

4. Originally Posted by mastermin346
The first and third term of arithmetic progression are x and y respectively.Find the sixteenth term in term of x and y.
$y = x + 2d \Rightarrow d = \frac{y - x}{2}$. And the first term is x.

You now have everything necessary to do anything you want, including finding the sixteenth term.