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Number Patterns

Chapter 7 Study Notes · O-Level 4052

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Section 6 of 7
Contents▾
7.0Why Number Patterns Matter7.1Patterns in Number Sequences7.2General Term of a Sequence7.3Finding the General Term of a Linear Sequence7.4Finding the General Term by Comparison7.5Patterns in Geometry and Real-World Contexts7.6Summary and Cross-References

7.5Patterns in Geometry and Real-World Contexts

Counting objects in a growing figure, or quantities in a real situation, gives a sequence. The method never changes.

  1. Count the objects in each figure and tabulate nn against TnT_n.
  2. Check whether the differences are constant.
  3. Use the formula for "how many in figure kk" and "is there a figure with exactly mm".
Figure 1:4 sticksFigure 2:7 sticksFigure 3:10 stickseach step adds the three gold sticks, so the difference is 3
The matchstick squares from the worked example. Each step adds three sticks, and the very first upright is the +1+1.

Figure sequence: matchstick squares

Matchstick squares in a row: Figure 1 uses 4 sticks, Figure 2 uses 7, Figure 3 uses 10. Find (i) Sn\displaystyle S_n, (ii) Figure 50, (iii) whether any figure uses exactly 100.

Show solution▾
(i) Common difference d=7−4d = 7 - 4
=3\displaystyle = 3
Sn\displaystyle S_n
=3n+c\displaystyle = 3n + c
c\displaystyle c
=S1−d\displaystyle = S_1 - d
=4−3\displaystyle = 4 - 3
=1\displaystyle = 1
Sn\displaystyle S_n
=3n+1\displaystyle = 3n + 1
Check: S1=3(1)+1=4S_1 = 3(1) + 1 = 4 ✓, S4=3(4)+1=13S_4 = 3(4) + 1 = 13 ✓
(ii) S50=3(50)+1S_{50} = 3(50) + 1
=150+1\displaystyle = 150 + 1
=151 sticks\displaystyle = 151 \text{ sticks}
(iii) Set 3n+1=1003n + 1 = 100
3n\displaystyle 3n
=99\displaystyle = 99
n\displaystyle n
=33\displaystyle = 33
33 is a positive integer, so Figure 33 uses exactly 100 sticks.

Real-world pattern: stacking tins

A stack of tins has 20 in the bottom row, 18 in the next, then 16. Find (i) Tn\displaystyle T_n for the n\displaystyle nth row up, (ii) row 8, (iii) which row holds exactly 12.

Show solution▾
(i) Common difference d=18−20d = 18 - 20
=−2\displaystyle = -2
Tn\displaystyle T_n
=−2n+T0\displaystyle = -2n + T_0
T0\displaystyle T_0
=T1−d\displaystyle = T_1 - d
=20−(−2)\displaystyle = 20 - (-2)
=22\displaystyle = 22
Tn\displaystyle T_n
=−2n+22\displaystyle = -2n + 22
Check: T1=−2(1)+22=20T_1 = -2(1) + 22 = 20 ✓, T2=−2(2)+22=18T_2 = -2(2) + 22 = 18 ✓
(ii) T8=−2(8)+22T_8 = -2(8) + 22
=−16+22\displaystyle = -16 + 22
=6 tins\displaystyle = 6 \text{ tins}
(iii) Set −2n+22=12-2n + 22 = 12
−2n\displaystyle -2n
=−10\displaystyle = -10
n\displaystyle n
=5\displaystyle = 5
The 5th row has exactly 12 tins.
  • The Fibonacci sequence 1,1,2,3,5,8,13,21,34,55,…1, 1, 2, 3, 5, 8, 13, 21, 34, 55, \ldots adds the two terms before it.
  • Consecutive ratios approach the Golden Ratio, about 1.6181.618.

Now you try

Paving slabs are laid around a growing pond. Design 1 uses 66 slabs, Design 2 uses 1111 and Design 3 uses 1616. Find SnS_n, the number of slabs in Design nn.

Show solution▾
Design 1: 66, Design 2: 1111, Design 3: 1616
Common difference d\displaystyle \text{Common difference } d
=11−6\displaystyle = 11 - 6
=5\displaystyle = 5
Sn\displaystyle S_n
=5n+c\displaystyle = 5n + c
c\displaystyle c
=S1−d\displaystyle = S_1 - d
=6−5\displaystyle = 6 - 5
=1\displaystyle = 1
Sn\displaystyle S_n
=5n+1\displaystyle = 5n + 1
Check: S3=5(3)+1=16S_3 = 5(3) + 1 = 16, which is what Design 3 uses.

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