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9.2.2 Challenging number sequences

For number patterns that are not directly related to its position in the sequence,
we can compare it with a known number sequence, like the multiples of a
number, perfect squares or cubes.

Example 3

Find the formula for the general term (Tn) of each sequence.
O 4,7, 10, 13, 16,...
@ 1, 6, 11, 16, 21, ...

Solution
o
Find the differences between the terms.

Position n 1 2 3 4 5
TermTn 4, 7, 10, 13, 16,

We start at 4 and add 3 each time.
We say the common difference = 3.

Because the common difference is 3, we compare the given sequence to
the multiples of 3.

Position in the sequence, n 1 2 3 4 5
Terms 4 7 10 13 16
Multiples of 3 3 6 9 12 15

1
1
3 + 6 + 1 9 + 12+1 15 + 1
1
(Multiples of 3) +
= 4 = 7 = 10 = 13 = 16
So, each term in the given sequence = multiples of 3 + 1.
Ti = 4 = 3 + 1= 3x1 + 1
T2 = 7 = 6 + 1= 3x2 + 1
T3 = 10 = 9 + 1= 3x3 + 1
T4 = 13 = 12 + 1 =3 X 4 + 1
T5 = 16 = 15 + 1 =3 X 5 +
1
So, Tn = 3 X /! + 1 = 3/7 + 1.
©

Find the differences between the terms.
1
Position n 1 2 3 4 5
Term In 1, 6, 11, 16, 21, ...

+ 5 +5 +5 +5
We start at 1 and add 5 each time.
We say the common difference = 5.

UIMIT 9 I Terms and Sequences