Sequences and Series

Lesson

When we have a graphing calculator or similar technology available, it can make some of our work with Arithmetic Series far easier to analyse and calculate. Let's explore some of these options with a few examples.

The fifth term of an arithmetic sequence is $18$18 and the sum of the first ten terms is $190$190. Find the value of the first term and the common difference.

Think: To find $a$`a` and $d$`d` we have two pieces of information and could therefore establish two equations and solve them algebraically. However, since our series formula is a quadratic and our sequences formula is linear, this is a bit tricky. With a CAS calculator we can establish two equations and then solve them quickly.

Do:

Using our formula for an arithmetic sequence we have: $18=a+d\left(5-1\right)$18=`a`+`d`(5−1)

Using our formula for an arithmetic series we have: $190=\frac{10}{2}\left(2a+d\left(10-1\right)\right)$190=102(2`a`+`d`(10−1))

Now we can type these into the simultaneous solver on our calculator.

For the sequence $90$90, $85$85, $80$80, $75$75 ... determine:

(a) When the sum is first greater than $400$400.

Think: With the sequence facility of our calculator we can quickly and easily scroll through the terms of the sequence and the series to find when the sum is first greater than $400$400. Firstly we'll need a rule to define this sequence.

Do: We can define this sequence by $T_n=90-5\left(n-1\right)$`T``n`=90−5(`n`−1)

Putting this in our calculator, we can now search for a sum greater than $400$400.

So we can see that after $6$6 terms of the sequence the sum is greater than $400$400.

(b) When the sum is zero.

Think: We will keep scrolling through the terms of our series in the third column to see when the sum of the terms reaches zero.

Do:

We can see that because the terms of our sequence start to become negative, the sum of the terms or our series decreases until at term $37$37 we have a sum of zero.

Find the sum of the first $20$20 terms of the arithmetic sequence defined by $T_1=\pi$`T`1=π and $T_{20}=20\pi$`T`20=20π.

Use the summation feature of a graphing calculator to find the sum of the first ten terms, $S_{10}$`S`10, of the arithmetic series defined by $T_n=-\sqrt[3]{5}n+\sqrt{3}$`T``n`=−^{3}√5`n`+√3.

Give your answer to the nearest thousandth.

The $17$17th term of an arithmetic sequence is $-307$−307.

The sum of the first $12$12 terms of this arithmetic sequence is $-534$−534.

We want to find the first term and common difference of this sequence.

Using the value of the $17$17th term of this arithmetic sequence, write an equation involving $a$

`a`, the first term, and $d$`d`, the common difference.Write your answer such that the constant term is by itself on one side of the equation.

Using value of the sum of the first $12$12 terms of this arithmetic sequence, write another equation involving $a$

`a`, the first term, and $d$`d`, the common differenceWrite your answer such that the constant term is by itself on one side of the equation.

Use the simultaneous solving facility of your calculator to determine the values of $a$

`a`and $d$`d`.$a$ `a`$=$= $\editable{}$ $d$ `d`$=$= $\editable{}$

Use arithmetic and geometric sequences and series

Apply sequences and series in solving problems