The sum of the 4 th and 6 th terms of an
arithmetical progression is 42 . The sum of
the 3rd and 9th terms of the progression is
52 . Find the first term, the common
difference and the sum of the first ten

Let’s start with a fun fact: the terms of an arithmetic progression (AP) can be represented mathematically. If ‘a’ is the first term and ‘d’ is the common difference, the nth term can be expressed as . This means that the 4th term is and the 6th term is .

Now onto the mysteries of AP! From your clues, we know:

Now you can solve these simultaneous equations:

From the equations and , subtracting the first from the second gives . Now substituting back into the first equation yields or .

To find the sum of the first ten terms, you can use the formula: . For ten terms:

.

So, the first term is 1, the common difference is 5, and the sum of the first ten terms is 235!