Sequences [27 marks]
1. [Maximum mark: 27]
This question asks you to explore some properties of polygonal numbers
and to determine and prove interesting results involving these numbers.
A polygonal number is an integer which can be represented as a series of dots
arranged in the shape of a regular polygon. Triangular numbers, square numbers
and pentagonal numbers are examples of polygonal numbers.
For example, a triangular number is a number that can be arranged in the shape
of an equilateral triangle. The first five triangular numbers are 1, 3, 6, 10 and
15.
The following table illustrates the first five triangular, square and pentagonal
numbers respectively. In each case the first polygonal number is one represented
by a single dot.
For an r-sided regular polygon, where r ∈ Z
+
, r ≥ 3, the nth polygonal
number P r (n) is given by
2
(r−2)n −(r−4)n
P r (n) =
2
, where n ∈ Z
+
.
2
(4−2)n −(4−4)n
Hence, for square numbers, P 4 (n) =
2
= n
2
.
(a.i) For triangular numbers, verify that P 3 (n) =
n(n+1)
. [2]
2
(a.ii) The number 351 is a triangular number. Determine which one
it is. [2]
(b.i) Show that P 3 (n) + P 3 (n + 1) ≡ (n + 1)
2
. [2]
(b.ii) State, in words, what the identity given in part (b)(i) shows for
two consecutive triangular numbers. [1]
(b.iii) For n = 4, sketch a diagram clearly showing your answer to
part (b)(ii). [1]
(c) Show that 8P 3 (n) + 1 is the square of an odd number for all
n ∈ Z
+
. [3]
The nth pentagonal number can be represented by the arithmetic series
P 5 (n) = 1 + 4 + 7 + … + (3n − 2).
(d) Hence show that P 5 (n) =
n(3n−1)
for n ∈ Z
+
. [3]
2
(e) By using a suitable table of values or otherwise, determine the
smallest positive integer, greater than 1, that is both a
triangular number and a pentagonal number. [5]
(f ) A polygonal number, P r (n), can be represented by the series
Σ (1 + (m − 1)(r − 2)) where r ∈ Z , r ≥ 3.
+
m=1
Use mathematical induction to prove that
2
(r−2)n −(r−4)n
P r (n) =
2
where n ∈ Z
+
. [8]
© International Baccalaureate Organization, 2024
