Video Transcript
Find in terms of π the sum of the
arithmetic sequence nine, 10, 11, and so on up to π plus eight.
We know that we can calculate the
sum of any arithmetic sequence using one of two formulae. Firstly, π sub π is equal to π
over two multiplied by two π plus π minus one multiplied by π. In this formula, π is the first
term and π is the common difference. Alternatively, π sub π is equal
to π over two multiplied by π plus π, where π is the first term and π is the
last term.
We can see from our sequence that
the first term π is nine and the common difference is one as the numbers are
increasing by one. The value of our last term π is π
plus eight. Substituting in our values of π
and π to the first formula gives us π over two multiplied by two multiplied by
nine plus π minus one multiplied by one. This simplifies to π over two
multiplied by 18 plus π minus one. We can simplify the square bracket
further such that the sum of the first π terms is equal to π over two multiplied
by π plus 17.
If we chose to use the second
formula, we need to substitute π equals nine and π equals π plus eight. This gives us π sub π is equal to
π over two multiplied by nine plus π plus eight. Once again, the bracket or
parentheses simplifies to π plus 17.
The sum of the arithmetic sequence
in terms of π is π over two multiplied by π plus 17.