Sequence of Numbers
Sequence of Numbers
Mun Chou, Fong
QED Education Scientific
Malaysia
LEVEL
High school after students have learned sequence.
OBJECTIVES
To review sequences and generate sequences using scientific calculator.
OVERVIEW
We briefly review properties of sequence of numbers; following the review we explore
about using scientific calculator to generate the sequence from different approaches. In
this exploration the calculator in discussion is the fx-991ES.
EXPLORATORY ACTIVITIES
[Note]
(a) We shall use small letter x an y instead of capitals X and Y as shown on the
calculator throughout the paper.
(b) Unless otherwise specified, we choose MthIO mode in the SETUP menu by tapping
Sequence in Review
Sequence is a function by definition. We usually deal either with finite or infinite sequence.
For example, the sequence of numbers 1, 3, 5, 7, 9 is a finite sequence; while the
function f (n) = n + 1 , for n ∈ positive integer set, is an infinite sequence.
Some other finite and infinite sequences:
(i) Prime numbers: 2, 3, 5, 7, 11, 13, 17,… is an infinite sequence.
(ii) All positive odd number less then 20: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19 is finite
(iii) Fibonacci numbers: 1, 1, 2, 3, 5, 8, 13, 21, 34, … is infinite.
Usually, the terms within the sequence can be described by some pattern and as such the
sequence can be represented with a general term, although some sequences do not
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Sequence of Numbers
observe any general patterns, such as the sequence of prime numbers.
In this exploration we denote the general term as Tn. In the study of sequence we are
usually required to do one or all of the following: generate the sequence, find the general
term, find the n-th term and calculate the sum of the first n-th term. For illustration let’s
consider this example.
Example 1: Given the sequence of all positive even numbers,
(i) Write down the first 7 terms of the sequence.
To generate the infinite sequence is fairly easy: 2, 4, 6, 8, 10, 12, 14,…
(ii) Find the general term of the sequence.
The general term can be represented as such: Tn = 2n, where n ∈ positive integer set.
(iii) Find the 15th term.
Using the general term, we have T15 = 2(15) = 30.
(iv) Find the sum of the first 12 term.
The sum of the first n-th term is denoted as Sn in this exploration. We are required to find
S12 = 2+4+6+8+10+12+14+16+18+20+22+24 = 156.
As this example demonstrated, some calculations can be quite tedious. We now explore
the use of calculators to speed up the calculations. We also explore the possibility of using
the calculator to find the n-th term and the sum, with the general term Tn given.
Generate the Sequence with Table
Using the calculator we can generate the sequence much more efficiently. One good way
is by the use of Table.
n(n + 1) 2
Example 2: The n-th term of a sequence is Tn =
. Write down the first 15 terms.
2
Using the Table mode, we can generate these 15 terms efficiently. The Start value is 1
and End value is 15 while step size is always 1 for sequence. In the calculator we use x to
replace the n given in the general term.
[Operations]
• Enter Table mode
x ( x + 1) 2
• Now input f ( x ) =
2
x ( x + 1) 2
• Continue to input f ( x ) =
2
• Enter the Start and End values
• Followed by the Step size
A table of values will be tabulated from this function and the values on the ‘F(X)’ column
n(n + 1) 2
are the first 15 terms for the sequence represented by Tn =
.
2
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Sequence of Numbers
We can also find the n-th term using the table if we input the setting such that n is Start
value ≤ n ≤ End value of the table.
Sequence with Solver
A much better approach if we are to find the n-th term is through the use of the calculator
solve features as this feature is more accessible than the Table mode.
Example 3: Find the arithmetic mean of the 9th, 49th and the 73rd terms of the sequence
with the general term
Tn = 3n 3 − 4n 2 + 8 .
T9 + T49 + T73
and therefore the solution to this problem is not too
3
hard but the calculation is tedious. With the combination of the CALC function and the
independent memory, the problem can be calculated quite efficiently. Here again we
replace n in the general term with x.
We are required to find
[Operations]
• Enter Comp mode
• Clear the independent memory
• Now input 3x 3 − 4 x 2 + 8
• Continue to input 3x 3 − 4 x 2 + 8
• Now calculate T9
This gives T9 = 1871. Now store this value into the independent memory M.
• Store T9 into M
• Now scroll up to find T49
• Add T49 to T9 stored in M
• Scroll up to find T73
• Add T73 to sum in M
T + T49 + T73
• Now find 9
3
1490965
.
3
This example shows that proper use of technology such as a scientific calculator can help
to approach a problem efficiently and effectively.
Hence the arithmetic mean is
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Sequence of Numbers
Recursive Sequence with Answer Memory
Sequence where its n-th term is the function of previous terms is a recursive sequence.
The first term of a recursive sequence is usually given. We look at how we may utilize the
Answer Memory, or Ans, to help us solve problems on recursive sequence.
Example 4: The first term of the recursive sequence Tn = Tn2−1 + 3Tn−1 is given as 2. Find
the first 5 terms of the sequence.
From the calculation perspective the generation of the 5 terms is not difficult but tedious.
We can improve the calculation using the Answer Memory as follow. We key in the
expression Tn = Tn2−1 + 3Tn−1 but with the (n-1)-th term replaced by Ans.
[Operations]
• Enter Comp mode
• Store 2 into the Answer Memory
• Now enter Ans 2 + 3Ans
• Now calculate T2
actually calculates (2)2 + 3(2)
T1 is given as 2 in the problem, and our first press of
= 10, and this is the second term T2. We can find the next 3 terms by simply tapping on
the equal key.
[Operations]
• To calculate T3
• To calculate T4
• To calculate T5
The operation shows that the first 5 term of the recursive sequence as 2, 10, 130, 17290,
298995970.
Sum of the First n-th Terms
Finding sum of the first n-th term, or Sn, is important in the study of sequence. While
calculating the sum for a recursive sequence and sequence which has no general pattern
demands further study, finding the sum is usually quite academic though again the
working involved is tedious. The Summation feature of the 991-ES provides us with
avenue to improve this.
Example 5: Find the sum of the first 23 terms, or S23, for the following sequences.
(i) The sequence of numbers with general term Tn = 2n -1, for n = 1, 2, 3, 4, …
n(n + 1) 2
(ii) The sequence described in Example 2 whose general term is Tn =
.
2
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Sequence of Numbers
The Summation feature
∑( )
requires us to input the start and end values for the
x=
summation. The step size, or increment, within the Summation is fixed as 1. Hence in
using the Summation feature to solve (i), the operation is as follow.
[Operations]
• Enter Comp mode
• Enter the Summation mode
• Move cursor to start value box
• Enter 1 when the box is selected
• Now move up to end value box
• Enter 23 into the end value box
• Scroll to the function term box
Using x to replace n in Tn = 2n -1, enter the general term.
• Enter 2x -1
• To evaluate
The result is 529, and apparently the sequence is the sequence of positive odd number.
This result means 1 + 3 + 5 + 7 + 9 … + 41 + 43 + 45 = 529.
To solve (ii) the operation is as followed.
[Operations]
• Enter Comp mode
• Enter the Summation mode
• Move cursor to start value box
• Enter 1 when the box is selected
• Now move up to end value box
• Enter 23 into the end value box
• Scroll to the function term box
n(n + 1) 2
• Enter
2
• Continue entering then evaluate
The result is 42550 as shown on the calculator.
The summations of sequence, or Sn, are usually referred to as Series.
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Sequence of Numbers
EXERCISES
Use the few features discussed above to solve these exercises. You can choose to work on
the same problems with different features or a combination of them.
Exercise 1
Write down the 5th till 19th term of the sequence with general term Tn = 5 − 2n 2 .
Exercise 2
Find the first 14 terms of the sequence with general term Tn = 7n 3 + n − 2n 2 . Then
calculate the sum of its first 14 terms.
Exercise 3
Find the geometric mean of the 5th, 7th and the 10th terms of the sequence with the
general term
5n
Tn = 2
n +1
Hint: Geometric mean of 3 numbers α, β and γ is
3
αβγ .
Exercise 4
The first term of the recursive sequence Tn = 2Tn2−1 − 1 is given as 5. Find the sum of the
first 7 terms of the sequence.
Exercise 5
Find the sum of the first 100 terms for the sequences (i) Tn = n2 and (ii) Tn =
1
.
n2
SOLUTIONS to Exercises
Exercise 1
-45, -67, -93, -123, -157, -195, -237, -283, -333, -387, -445, 507, -573, -643, -717
Exercise 2
6, 50, 174, 420, 830, 1446, 2310, 3464, 4950, 6810, 9086, 11820, 15054, 18830.
Sum =75250.
Exercise 3
Geometric mean is 0.693273 correct to 6 decimal places.
Exercise 4
Approximately 2.6107 X 1063.
Exercise 5
(i) 338350 and (ii) 1.635.
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