### Video Transcript

Find an expression for the sum of
an arithmetic sequence whose first term is π and whose common difference is π.

We are told in the question that
the first term of our arithmetic sequence is π and the common difference is π. We are trying to find an expression
for the sum of the first π terms which we will write as π sub π. This will be equal to the first
term π plus the second term π plus π and so on. We know that the πth term of any
arithmetic sequence is equal to π plus π minus one multiplied by π. This means that the penultimate
term is equal to π plus π minus two multiplied by π. We will call this equation one.

We will then reverse the order of
this sum, which we can do as addition is commutative. This gives us π sub π is equal to
π plus π minus one multiplied by π plus π plus π minus two multiplied by π and
so on and finally plus π plus π plus π. We will call this equation two. Adding equation one and equation
two gives us two multiplied by π sub π on the left-hand side. On the right-hand side, we will add
each pair of terms. Adding the first pair, we see that
π plus π is equal to two π. So we have two π plus π minus one
multiplied by π.

The second pair of terms have the
same sum as π plus π is equal to two π and π plus π minus two π is equal to π
minus one π. In fact, this will be true for each
of the pairs in our equations. Each pair of terms will sum to give
us two π plus π minus one multiplied by π. We have π of these terms, so we
can rewrite the right-hand side as π multiplied by two π plus π minus one
multiplied by π. Dividing both sides of our equation
by two gives us π sub π is equal to π over two multiplied by two π plus π minus
one multiplied by π. This is an expression for the sum
of an arithmetic sequence whose first term is π and whose common difference is
π.